JEE Advanced Algebra Cheat Sheet — Complex Numbers, Sequences, Matrices

Cracking JEE Advanced requires a deep understanding of core concepts and the ability to apply them efficiently. This comprehensive cheat sheet provides a concise yet thorough review of Complex Numbers, Sequences & Series, and Matrices & Determinants – crucial topics in JEE Advanced Algebra. Master these areas, and you'll significantly boost your score.

JEE Advanced Algebra Cheat Sheet: Complex Numbers, Sequences, Matrices

Introduction to Algebra in JEE Advanced

Algebra forms the backbone of quantitative aptitude and is a consistently high-weightage section in JEE Advanced Mathematics. A strong grasp of Complex Numbers, Sequences & Series, and Matrices & Determinants is non-negotiable for success. These topics often intertwine with calculus, coordinate geometry, and vector algebra, demanding both conceptual clarity and strong problem-solving skills. This resource aims to distill the essential formulas, properties, and problem-solving strategies into an easily digestible format.

Exam Weightage and Trends

Historically, Algebra contributes approximately 30-40% of the total marks in JEE Advanced Mathematics. Within Algebra:

• Complex Numbers: Typically 1-2 questions, sometimes integrated with coordinate geometry or trigonometry. Can be high difficulty.
• Sequences & Series: Often 1-2 questions, frequently involving sum to infinity, properties of AP/GP/HP, or combined series. Arithmetic-Geometric Progression (AGP) and special series sums are common.
• Matrices & Determinants: Consistently 2-3 questions. System of linear equations, inverse, adjoint, properties of determinants, rank, eigenvalues (though less common in JEE directly) are key areas. Often conceptual and calculation-intensive.

Recent trends suggest increasing emphasis on conceptual understanding over rote memorization, and problems that require combining multiple concepts from different chapters.

1. Complex Numbers

Complex numbers extend the real number system to include the imaginary unit $i$, where $i^2 = -1$. They are fundamental for solving many polynomial equations and have wide applications in physics and engineering.

1.1. Basic Definitions and Operations

• Definition: A complex number $z$ is expressed as $z = x + iy$, where $x, y \in \mathbb{R}$ and $i = \sqrt{-1}$.
  • $x = \text{Re}(z)$ (real part)
  • $y = \text{Im}(z)$ (imaginary part)
• Conjugate: The conjugate of $z = x + iy$ is $\bar{z} = x - iy$.
  • Properties:
    • $z + \bar{z} = 2 \text{Re}(z)$
    • $z - \bar{z} = 2i \text{Im}(z)$
    • $z \bar{z} = x^2 + y^2 = |z|^2$
    • $\overline{z_1 \pm z_2} = \bar{z_1} \pm \bar{z_2}$
    • $\overline{z_1 z_2} = \bar{z_1} \bar{z_2}$
    • $\overline{z_1 / z_2} = \bar{z_1} / \bar{z_2}$
    • $\overline{z^n} = (\bar{z})^n$
• Modulus (Magnitude): The modulus of $z = x + iy$ is $|z| = \sqrt{x^2 + y^2}$. Geometrically, it's the distance from the origin to the point $(x,y)$ in the Argand plane.
  • Properties:
    • $|z| \ge 0$, $|z|=0 \iff z=0$
    • $|z_1 z_2| = |z_1| |z_2|$
    • $|z_1 / z_2| = |z_1| / |z_2|$
    • $|z^n| = |z|^n$
    • $|z_1 + z_2| \le |z_1| + |z_2|$ (Triangle Inequality)
    • $|z_1 - z_2| \ge ||z_1| - |z_2||$ (Reverse Triangle Inequality)
    • $\text{Re}(z) \le |z|$, $\text{Im}(z) \le |z|$
    • $|z_1 + z_2|^2 = (z_1+z_2)(\bar{z_1}+\bar{z_2}) = |z_1|^2 + |z_2|^2 + z_1\bar{z_2} + \bar{z_1}z_2 = |z_1|^2 + |z_2|^2 + 2\text{Re}(z_1\bar{z_2})$
• Argument (Amplitude): The principal argument of $z = x + iy$ is $\text{arg}(z) = \theta$, where $x = |z|\cos\theta$ and $y = |z|\sin\theta$.
  • $-\pi < \theta \le \pi$
  • $\text{arg}(z) = \tan^{-1}(y/x)$, taking into account the quadrant of $(x,y)$.
  • If $x=0$, $y>0$, $\text{arg}(z) = \pi/2$. If $x=0$, $y<0$, $\text{arg}(z) = -\pi/2$.
  • Properties:
    • $\text{arg}(z_1 z_2) = \text{arg}(z_1) + \text{arg}(z_2) + 2k\pi$ (for some integer $k$)
    • $\text{arg}(z_1 / z_2) = \text{arg}(z_1) - \text{arg}(z_2) + 2k\pi$
    • $\text{arg}(z^n) = n \text{arg}(z) + 2k\pi$
    • $\text{arg}(\bar{z}) = -\text{arg}(z)$

1.2. Forms of Complex Numbers

• Cartesian (Rectangular) Form: $z = x + iy$
• Polar Form: $z = r(\cos\theta + i\sin\theta)$, where $r = |z|$ and $\theta = \text{arg}(z)$.
• Euler's Form: $z = re^{i\theta}$, where $e^{i\theta} = \cos\theta + i\sin\theta$.

1.3. De Moivre's Theorem

For any integer $n$: $$(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)$$ In Euler's form: $(e^{i\theta})^n = e^{in\theta}$. This also extends to rational powers $(p/q)$ for finding roots.

1.4. Roots of Unity

$n$-th roots of unity: Solutions to $z^n = 1$.

• These are $z_k = e^{i (2k\pi/n)} = \cos(2k\pi/n) + i\sin(2k\pi/n)$ for $k = 0, 1, \dots, n-1$.
• These roots form the vertices of a regular $n$-gon inscribed in the unit circle centered at the origin in the Argand plane.
• Properties:
  • Sum of roots: $\sum_{k=0}^{n-1} z_k = 0$ (if $n > 1$)
  • Product of roots: $\prod_{k=0}^{n-1} z_k = (-1)^{n-1}$
  • If $\omega$ is an $n$-th root of unity (other than 1), then $1 + \omega + \omega^2 + \dots + \omega^{n-1} = 0$.

Cube Roots of Unity: Solutions to $z^3 = 1$.

• $z = 1, \omega, \omega^2$
  • $\omega = e^{i(2\pi/3)} = \cos(2\pi/3) + i\sin(2\pi/3) = -1/2 + i\sqrt{3}/2$
  • $\omega^2 = e^{i(4\pi/3)} = \cos(4\pi/3) + i\sin(4\pi/3) = -1/2 - i\sqrt{3}/2$
• Properties:
  • $1 + \omega + \omega^2 = 0$
  • $\omega^3 = 1$
  • $\omega^2 = \bar{\omega} = 1/\omega$

1.5. Geometry of Complex Numbers

• Point: A complex number $z = x+iy$ represents a point $(x,y)$ in the Argand plane.
• Vector: $z$ can be seen as a vector from the origin to $(x,y)$.
• Distance: $|z_1 - z_2|$ is the distance between points $z_1$ and $z_2$.
• Section Formula: If $P(z)$ divides the line segment joining $A(z_1)$ and $B(z_2)$ in the ratio $m:n$, then $z = \frac{nz_1 + mz_2}{m+n}$. For midpoint, $z = \frac{z_1+z_2}{2}$.
• Rotation: Multiplication of $z$ by $e^{i\alpha}$ rotates $z$ about the origin by an angle $\alpha$ counter-clockwise.
  • To rotate $z_1$ about $z_0$ by angle $\alpha$: $z - z_0 = (z_1 - z_0)e^{i\alpha}$.
• Loci:
  • $|z - z_0| = r$: Circle with center $z_0$ and radius $r$.
  • $|z - z_1| = |z - z_2|$: Perpendicular bisector of the line segment joining $z_1$ and $z_2$.
  • $\text{arg}(z - z_0) = \alpha$: Ray starting from $z_0$ making an angle $\alpha$ with the positive real axis.
  • $\text{arg}\left(\frac{z-z_1}{z-z_2}\right) = \alpha$:
    • If $\alpha = 0$ or $\pi$: Line passing through $z_1, z_2$.
    • If $\alpha \ne 0, \pi$: Arc of a circle passing through $z_1, z_2$.
  • $|z - z_1| + |z - z_2| = k$:
    • If $k = |z_1 - z_2|$, line segment joining $z_1, z_2$.
    • If $k > |z_1 - z_2|$, ellipse with foci $z_1, z_2$.
  • $||z - z_1| - |z - z_2|| = k$:
    • If $k = |z_1 - z_2|$, line segment joining $z_1, z_2$ (not including endpoints).
    • If $k < |z_1 - z_2|$, hyperbola with foci $z_1, z_2$.

1.6. Square Roots of a Complex Number

The square roots of $z = x+iy$ are $\pm \left( \sqrt{\frac{|z|+x}{2}} + i (\text{sgn}(y)) \sqrt{\frac{|z|-x}{2}} \right)$. Where $\text{sgn}(y) = 1$ if $y \ge 0$, and $-1$ if $y < 0$.

Example 1 (Complex Numbers):

If $z$ is a complex number such that $|z|=1$ and $z\ne \pm 1$, then find the value of $\frac{z-1}{z+1}$.

Solution: Let $w = \frac{z-1}{z+1}$. We need to determine the nature of $w$. Given $|z|=1$, we know $z\bar{z}=|z|^2=1$, so $\bar{z} = 1/z$.

Let's find $\bar{w}$: $\bar{w} = \overline{\left(\frac{z-1}{z+1}\right)} = \frac{\bar{z}-1}{\bar{z}+1}$ Substitute $\bar{z} = 1/z$: $\bar{w} = \frac{1/z - 1}{1/z + 1} = \frac{(1-z)/z}{(1+z)/z} = \frac{1-z}{1+z} = -\left(\frac{z-1}{z+1}\right) = -w$

So, $\bar{w} = -w$. Let $w = a+ib$. Then $\bar{w} = a-ib$. $a-ib = -(a+ib) \implies a-ib = -a-ib \implies 2a = 0 \implies a = 0$. This means $w$ is purely imaginary. Since $z \ne \pm 1$, $z-1 \ne 0$ and $z+1 \ne 0$. Thus, $w$ is a non-zero purely imaginary number.

Answer: $\frac{z-1}{z+1}$ is a purely imaginary number.

Common Mistakes in Complex Numbers:

• Assuming $\text{arg}(z_1 z_2) = \text{arg}(z_1) + \text{arg}(z_2)$ directly without considering the $2k\pi$ multiple to bring it into the principal argument range.
• Incorrectly applying triangle inequality or its reverse direction.
• Errors in calculating principal argument based on the quadrant.
• Forgetting that $\sqrt{ab} = \sqrt{a}\sqrt{b}$ is only true if at least one of $a,b$ is non-negative. For complex numbers, $\sqrt{-a}\sqrt{-b} \ne \sqrt{ab}$ (e.g., $\sqrt{-1}\sqrt{-1} = i \cdot i = -1 \ne \sqrt{1} = 1$).
• Assuming $\text{Im}(z) = iy$; it's $y$.

2. Sequences and Series

This section deals with ordered lists of numbers (sequences) and their sums (series), focusing on Arithmetic Progressions (AP), Geometric Progressions (GP), Harmonic Progressions (HP), and special series.

2.1. Arithmetic Progression (AP)

A sequence $a_1, a_2, \dots, a_n, \dots$ is an AP if $a_{k+1} - a_k = d$ (common difference) for all $k \ge 1$.

• $n$-th term: $a_n = a_1 + (n-1)d$
• Sum of $n$ terms: $S_n = \frac{n}{2}(a_1 + a_n) = \frac{n}{2}(2a_1 + (n-1)d)$
• Properties:
  • If $a,b,c$ are in AP, then $2b = a+c$.
  • If $a_1, \dots, a_n$ is an AP, then $a_k = \frac{a_{k-1} + a_{k+1}}{2}$ for $1 < k < n$.
  • $a_m = \frac{a_{m-k} + a_{m+k}}{2}$.
  • The sum of terms equidistant from the beginning and end is constant: $a_1 + a_n = a_2 + a_{n-1} = \dots$.
  • If terms are chosen at regular intervals from an AP, they form an AP.
  • If $a_1, \dots, a_n$ is an AP, then $ca_1+k, \dots, ca_n+k$ is also an AP.

2.2. Geometric Progression (GP)

A sequence $a_1, a_2, \dots, a_n, \dots$ is a GP if $a_{k+1} / a_k = r$ (common ratio) for all $k \ge 1$.

• $n$-th term: $a_n = a_1 r^{n-1}$
• Sum of $n$ terms: $S_n = \frac{a_1(r^n - 1)}{r-1}$ (for $r \ne 1$). If $r=1$, $S_n = n a_1$.
• Sum to infinity: If $|r| < 1$, $S_\infty = \frac{a_1}{1-r}$.
• Properties:
  • If $a,b,c$ are in GP, then $b^2 = ac$.
  • If terms are chosen at regular intervals from a GP, they form a GP.
  • If $a_1, \dots, a_n$ is a GP, then $a_1^c, \dots, a_n^c$ is also a GP.

2.3. Harmonic Progression (HP)

A sequence $a_1, a_2, \dots, a_n, \dots$ is an HP if the reciprocals $1/a_1, 1/a_2, \dots, 1/a_n, \dots$ are in AP.

• No general formula for the sum of $n$ terms.
• $n$-th term: If $1/a_n = 1/a_1 + (n-1)d$, then $a_n = \frac{1}{(1/a_1) + (n-1)d}$.
• Property: If $a,b,c$ are in HP, then $b = \frac{2ac}{a+c}$ (harmonic mean).

2.4. Means

• Arithmetic Mean (AM): For $n$ numbers $x_1, \dots, x_n$, $\text{AM} = \frac{x_1 + \dots + x_n}{n}$.
• Geometric Mean (GM): For $n$ positive numbers $x_1, \dots, x_n$, $\text{GM} = (x_1 x_2 \dots x_n)^{1/n}$.
• Harmonic Mean (HM): For $n$ positive numbers $x_1, \dots, x_n$, $\text{HM} = \frac{n}{1/x_1 + \dots + 1/x_n}$.
• Relationship: For positive numbers, $\text{AM} \ge \text{GM} \ge \text{HM}$. Equality holds if and only if all numbers are equal.

2.5. Arithmetic-Geometric Progression (AGP)

A series whose terms are products of corresponding terms of an AP and a GP. Example: $S = a + (a+d)r + (a+2d)r^2 + \dots + (a+(n-1)d)r^{n-1}$. To find the sum: Multiply $S$ by $r$, shift terms, and subtract. $S = a + (a+d)r + (a+2d)r^2 + \dots + (a+(n-1)d)r^{n-1}$ $rS = \quad ar + (a+d)r^2 + \dots + (a+(n-2)d)r^{n-1} + (a+(n-1)d)r^n$ $(1-r)S = a + dr + dr^2 + \dots + dr^{n-1} - (a+(n-1)d)r^n$ $(1-r)S = a + dr(1+r+\dots+r^{n-2}) - (a+(n-1)d)r^n$ $(1-r)S = a + dr \frac{1-r^{n-1}}{1-r} - (a+(n-1)d)r^n$ For infinite AGP ($|r|<1$): $S_\infty = \frac{a}{1-r} + \frac{dr}{(1-r)^2}$.

2.6. Special Series

• Sum of first $n$ natural numbers: $\sum_{k=1}^n k = \frac{n(n+1)}{2}$
• Sum of squares of first $n$ natural numbers: $\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}$
• Sum of cubes of first $n$ natural numbers: $\sum_{k=1}^n k^3 = \left(\frac{n(n+1)}{2}\right)^2$
• Sum of terms of binomial expansion: $(1+x)^n = \binom{n}{0} + \binom{n}{1}x + \dots + \binom{n}{n}x^n$.
• Exponential series: $e^x = \sum_{n=0}^\infty \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots$
• Logarithmic series: $\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \dots$ for $-1 < x \le 1$.
• Taylor/Maculaurin Series: $f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n$.

2.7. Method of Differences

If the differences between consecutive terms of a sequence are in AP or GP, this method can be used to find the general term and sum. Let the sequence be $T_1, T_2, T_3, \dots, T_n$. First differences: $d_1 = T_2-T_1, d_2 = T_3-T_2, \dots$. If $d_1, d_2, \dots$ is an AP or GP, then $T_n = T_1 + \sum_{k=1}^{n-1} d_k$ (where $d_k$ is the $k$-th term of the difference sequence).

Example 2 (Sequences & Series):

The sum of an infinite geometric series with positive terms is $3$ and the sum of the cubes of its terms is $\frac{27}{19}$. Find the series.

Solution: Let the infinite GP be $a, ar, ar^2, \dots$. Since terms are positive, $a > 0$ and $r > 0$. The sum of the series is $S_\infty = \frac{a}{1-r} = 3$. (Equation 1) For this sum to exist, we must have $|r| < 1$, which is $0 < r < 1$ since $r>0$.

Now, consider the series of cubes of its terms: $a^3, (ar)^3, (ar^2)^3, \dots$, which is $a^3, a^3r^3, a^3r^6, \dots$. This is also an infinite GP with first term $A = a^3$ and common ratio $R = r^3$. Since $0 < r < 1$, we have $0 < r^3 < 1$, so this infinite sum also converges. The sum of this new series is $S'_\infty = \frac{a^3}{1-r^3} = \frac{27}{19}$. (Equation 2)

From Equation 1, $a = 3(1-r)$. Substitute this into Equation 2: $\frac{[3(1-r)]^3}{1-r^3} = \frac{27}{19}$ $\frac{27(1-r)^3}{1-r^3} = \frac{27}{19}$ Divide by 27: $\frac{(1-r)^3}{1-r^3} = \frac{1}{19}$ Recall the identity $1-r^3 = (1-r)(1+r+r^2)$: $\frac{(1-r)^3}{(1-r)(1+r+r^2)} = \frac{1}{19}$ $\frac{(1-r)^2}{1+r+r^2} = \frac{1}{19}$ $19(1-2r+r^2) = 1+r+r^2$ $19 - 38r + 19r^2 = 1 + r + r^2$ $18r^2 - 39r + 18 = 0$ Divide by 3: $6r^2 - 13r + 6 = 0$

We can solve this quadratic equation using the quadratic formula $r = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ or by factoring. $6r^2 - 9r - 4r + 6 = 0$ $3r(2r-3) - 2(2r-3) = 0$ $(3r-2)(2r-3) = 0$ So, $r = 2/3$ or $r = 3/2$.

Since we established $0 < r < 1$, we must choose $r = 2/3$. Now substitute $r=2/3$ back into $a = 3(1-r)$: $a = 3(1 - 2/3) = 3(1/3) = 1$.

The first term is $a=1$ and the common ratio is $r=2/3$. The series is $1, 1 \cdot (2/3), 1 \cdot (2/3)^2, \dots$ Series: $1, 2/3, 4/9, 8/27, \dots$

Answer: The series is $1, 2/3, 4/9, 8/27, \dots$.

Common Mistakes in Sequences and Series:

• Forgetting the condition $|r|<1$ for the sum to infinity of a GP.
• Incorrectly applying formulas for $S_n$ or $a_n$ (e.g., mixing AP and GP formulas).
• Errors in algebraic manipulation when solving for $a$ or $d/r$.
• Not checking for positive terms/conditions given in the problem statement (e.g. $a>0, r>0$).
• Confusion between AP, GP, and HP particularly when dealing with means.

3. Matrices and Determinants

Matrices are rectangular arrays of numbers, and determinants are scalar values associated with square matrices. Both are powerful tools for solving systems of linear equations, transformations, and other mathematical problems.

3.1. Matrices

3.1.1. Basic Definitions and Types

• Matrix: An ordered rectangular array of numbers (or functions). An $m \times n$ matrix has $m$ rows and $n$ columns.
• Order: The number of rows $\times$ the number of columns (e.g., $m \times n$).
• Types of Matrices:
  • Row Matrix: $1 \times n$ (one row).
  • Column Matrix: $m \times 1$ (one column).
  • Square Matrix: $m=n$.
  • Diagonal Matrix: Square matrix where all non-diagonal elements are zero.
  • Scalar Matrix: Diagonal matrix where all diagonal elements are equal.
  • Identity Matrix ($I$): Scalar matrix where diagonal elements are 1. $I = \begin{pmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{pmatrix}$.
  • Zero/Null Matrix ($O$): All elements are zero.
  • Upper Triangular Matrix: Elements below the main diagonal are zero.
  • Lower Triangular Matrix: Elements above the main diagonal are zero.
  • Symmetric Matrix: $A = A^T$ (where $A^T$ is the transpose of $A$).
  • Skew-Symmetric Matrix: $A = -A^T$. Diagonal elements must be zero.
  • Orthogonal Matrix: $AA^T = A^T A = I$. (Determinant is $\pm 1$).
  • Idempotent Matrix: $A^2 = A$.
  • Nilpotent Matrix: $A^k = O$ for some positive integer $k$. Smallest such $k$ is the index.
  • Involutory Matrix: $A^2 = I$.

3.1.2. Matrix Operations

• Addition/Subtraction: Only for matrices of the same order. $A \pm B = [a_{ij} \pm b_{ij}]$.
• Scalar Multiplication: $kA = [ka_{ij}]$.
• Matrix Multiplication: $AB$ is defined if the number of columns in $A$ equals the number of rows in $B$. If $A$ is $m \times n$ and $B$ is $n \times p$, then $AB$ is $m \times p$. Element $(AB){ij} = \sum{k=1}^n a_{ik}b_{kj}$.
  • Matrix multiplication is generally NOT commutative ($AB \ne BA$).
  • It IS associative: $(AB)C = A(BC)$.
  • It IS distributive: $A(B+C) = AB+AC$.
• Transpose ($A^T$ or $A'$): Rows become columns and columns become rows. $(A^T){ij} = A{ji}$.
  • Properties: $(A^T)^T = A$, $(A+B)^T = A^T+B^T$, $(kA)^T = kA^T$, $(AB)^T = B^T A^T$.

3.2. Determinants

A scalar value associated with a square matrix. Denoted as $\det(A)$ or $|A|$.

3.2.1. Calculation

• $2 \times 2$ matrix: If $A = \begin{pmatrix} a & b \ c & d \end{pmatrix}$, then $\det(A) = ad - bc$.
• $3 \times 3$ matrix: If $A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{pmatrix}$, then $\det(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})$.
  • Minor ($M_{ij}$): Determinant of the submatrix obtained by deleting $i$-th row and $j$-th column.
  • Cofactor ($C_{ij}$): $C_{ij} = (-1)^{i+j} M_{ij}$.
  • $\det(A) = \sum_{j=1}^n a_{ij}C_{ij}$ (expansion along $i$-th row) or $\det(A) = \sum_{i=1}^n a_{ij}C_{ij}$ (expansion along $j$-th column).

3.2.2. Properties of Determinants

• $\det(A^T) = \det(A)$.
• If two rows or two columns are interchanged, the sign of the determinant changes.
• If two rows or rows/columns are identical or proportional, $\det(A) = 0$.
• If a row or column consists entirely of zeros, $\det(A) = 0$.
• If all elements of a row or column are multiplied by a scalar $k$, the determinant is multiplied by $k$.
  • $\det(kA) = k^n \det(A)$ for an $n \times n$ matrix $A$.
• If each element of a row or column is expressed as a sum of two terms, the determinant can be expressed as a sum of two determinants.
• The value of a determinant remains unchanged if to any row (or column) we add $k$ times the corresponding elements of another row (or column). (Row/Column operations).
• $\det(AB) = \det(A)\det(B)$.
• For a diagonal matrix, $\det(A)$ is the product of its diagonal elements.
• If $A$ is triangular (upper or lower), $\det(A)$ is the product of its diagonal elements.
• Sum of products of elements of any row/column with cofactors of other row/column is zero: $\sum_{j=1}^n a_{ij}C_{kj} = 0$ if $i \ne k$.

3.2.3. Adjoint and Inverse of a Matrix

• Adjoint of A ($\text{adj}(A)$): Transpose of the cofactor matrix. If $C = [C_{ij}]$, then $\text{adj}(A) = C^T$.
• Properties of Adjoint:
  • $A (\text{adj}(A)) = (\text{adj}(A)) A = \det(A) I$.
  • $\det(\text{adj}(A)) = (\det(A))^{n-1}$ for an $n \times n$ matrix.
  • $\text{adj}(\text{adj}(A)) = (\det(A))^{n-2} A$.
  • $\det(\text{adj}(\text{adj}(A)) ) = (\det(A))^{(n-1)^2}$.
  • $(\text{adj}(AB)) = (\text{adj}(B)) (\text{adj}(A))$.
  • $(\text{adj}(A^T)) = (\text{adj}(A))^T$.
• Inverse of A ($A^{-1}$): A matrix $A$ is invertible if and only if $\det(A) \ne 0$.
  • $A^{-1} = \frac{1}{\det(A)} \text{adj}(A)$.
  • Properties of Inverse:
    • $(A^{-1})^{-1} = A$.
    • $(AB)^{-1} = B^{-1}A^{-1}$.
    • $(A^T)^{-1} = (A^{-1})^T$.
    • If $A$ is symmetric, $A^{-1}$ is symmetric.

3.3. System of Linear Equations

For a system $AX=B$, where $A$ is the coefficient matrix, $X$ is the variable matrix, and $B$ is the constant matrix.

• Cramer's Rule: (For $n$ equations in $n$ variables, if $\det(A) \ne 0$) $x_j = \frac{\det(A_j)}{\det(A)}$, where $A_j$ is the matrix obtained by replacing the $j$-th column of $A$ with $B$.

• Matrix Method:

  • If $\det(A) \ne 0$: $X = A^{-1}B$. Unique solution (consistent).
  • If $\det(A) = 0$:
    • If $(\text{adj}(A))B = O$: Infinitely many solutions (consistent).
    • If $(\text{adj}(A))B \ne O$: No solution (inconsistent).

• Homogeneous System ($AX=O$):

  • If $\det(A) \ne 0$: Only trivial solution ($X=O$).
  • If $\det(A) = 0$: Non-trivial solutions exist (infinitely many solutions).

Example 3 (Matrices & Determinants):

Let $A$ be a $3 \times 3$ matrix such that $\text{adj}(A) = \begin{pmatrix} 7 & -7 & 7 \ -3 & 4 & -3 \ 1 & -1 & 1 \end{pmatrix}$. Then find $\det(A)$ and $A^{-1}$.

Solution: We are given $\text{adj}(A)$. We know the property: $\det(\text{adj}(A)) = (\det(A))^{n-1}$, where $n$ is the order of the matrix. Here, $n=3$, so $\det(\text{adj}(A)) = (\det(A))^{3-1} = (\det(A))^2$.

Let's calculate $\det(\text{adj}(A))$. $\det(\text{adj}(A)) = \begin{vmatrix} 7 & -7 & 7 \ -3 & 4 & -3 \ 1 & -1 & 1 \end{vmatrix}$ We can observe that Column 1 and Column 3 are identical. Property of determinants: If two columns (or rows) are identical, the determinant is zero. So, $\det(\text{adj}(A)) = 0$.

Now, substituting this into the property: $(\det(A))^2 = 0$ Therefore, $\det(A) = 0$.

Next, we need to find $A^{-1}$. The formula for the inverse of a matrix $A$ is $A^{-1} = \frac{1}{\det(A)} \text{adj}(A)$. Since we found $\det(A) = 0$, the inverse $A^{-1}$ does not exist. A matrix with a determinant of zero is called a singular matrix, and singular matrices do not have an inverse.

Answer: $\det(A) = 0$, and $A^{-1}$ does not exist.

Common Mistakes in Matrices and Determinants:

• Assuming matrix multiplication is commutative.
• Errors in cofactor expansion, especially signs.
• Forgetting that $\det(kA) = k^n \det(A)$.
• Incorrectly applying determinant properties (e.g., sum of determinants, or identity for row operations).
• Confusing $A^{-1}$ not existing with "no solution" for matrix equations. For $AX=B$, if $\det(A)=0$, it means there might be no solution or infinitely many, not necessarily no solution.

Worked Examples

Let's work through more challenging examples that combine multiple concepts.

Example 4 (Complex Numbers - Locus):

If $z_1, z_2, z_3$ are vertices of an equilateral triangle in the Argand plane, show that $z_1^2 + z_2^2 + z_3^2 = z_1z_2 + z_2z_3 + z_3z_1$.

Solution: For an equilateral triangle, if we rotate one vertex about another by $60^\circ$ (or $-60^\circ$), we get the third vertex. Let's rotate $z_3$ about $z_1$ by $60^\circ$ to get $z_2$. This means $z_2 - z_1 = (z_3 - z_1)e^{i\pi/3}$. Also, rotate $z_1$ about $z_2$ by $60^\circ$ to get $z_3$. This means $z_3 - z_2 = (z_1 - z_2)e^{i\pi/3}$. Therefore, $\frac{z_2 - z_1}{z_3 - z_1} = \frac{z_3 - z_2}{z_1 - z_2}$. Let this common ratio be $\lambda = e^{i\pi/3}$. So, $\frac{z_2 - z_1}{z_3 - z_1} = \lambda$ and $\frac{z_3 - z_2}{z_1 - z_2} = \lambda$. Squaring the common ratio (which is not necessarily required for this proof, but a common property): $\lambda^2 = e^{i2\pi/3} = \omega$. However, a more direct algebraic relation for equilateral triangle vertices is: $(z_1 - z_2)^2 + (z_2 - z_3)^2 + (z_3 - z_1)^2 = 0$. This is generally for triangles with sides of equal length, not equilateral.

The correct property for an equilateral triangle by complex numbers relation is: $z_1^2 + z_2^2 + z_3^2 - z_1z_2 - z_2z_3 - z_3z_1 = 0$. This can be derived from the rotation property. Take rotation of $z_2$ about $z_1$ to $z_3$: $(z_3 - z_1) = (z_2 - z_1)e^{i\pi/3}$. Take rotation of $z_1$ about $z_2$ to $z_3$: $(z_3 - z_2) = (z_1 - z_2)e^{-i\pi/3}$. No, it's simpler: If $z_1, z_2, z_3$ form an equilateral triangle, then the ratio $(z_1-z_2)/(z_2-z_3)$ must be a rotation by $e^{i\pi/3}$ or $e^{-i\pi/3}$. Another common approach: The origin is the centroid or circumcenter if the sum is zero, which is not stated.

A widely used identity for equilateral triangle vertices $z_1, z_2, z_3$ is: $z_1 + \omega z_2 + \omega^2 z_3 = 0$ or $z_1 + \omega^2 z_2 + \omega z_3 = 0$, where $\omega = e^{i2\pi/3}$. Let's assume $z_1 + \omega z_2 + \omega^2 z_3 = 0$. Squaring this expression does not directly yield the desired result.

Let's use the property that $(z_1 - z_2)^2 = (z_1 - z_3)(z_2 - z_3)$ is not always correct. The condition for an equilateral triangle with vertices $A(z_1)$, $B(z_2)$, $C(z_3)$ is: $|z_1-z_2| = |z_2-z_3| = |z_3-z_1|$. Squaring these gives: $|z_1-z_2|^2 = |z_2-z_3|^2 = |z_3-z_1|^2$. $(z_1-z_2)(\bar{z_1}-\bar{z_2}) = (z_2-z_3)(\bar{z_2}-\bar{z_3}) = (z_3-z_1)(\bar{z_3}-\bar{z_1})$. This doesn't seem to lead algebraically to the identity easily.

Let's use the rotation form more effectively. For an equilateral triangle, we can take $z_1, z_2, z_3$ such that rotating $z_3$ about $z_1$ by $60^\circ$ gives $z_2$. So, $(z_2 - z_1) = (z_3 - z_1) e^{i\pi/3}$. (1) Also, rotating $z_1$ about $z_2$ by $60^\circ$ gives $z_3$. So, $(z_3 - z_2) = (z_1 - z_2) e^{i\pi/3}$. (2) From (1), we can write $z_2 - z_1 = \lambda(z_3 - z_1)$ where $\lambda = e^{i\pi/3}$. From (2), we can write $z_3 - z_2 = \lambda(z_1 - z_2)$. This implies $\frac{z_2 - z_1}{z_3 - z_1} = \frac{z_3 - z_2}{z_1 - z_2}$. Cross-multiply: $(z_2 - z_1)(z_1 - z_2) = (z_3 - z_1)(z_3 - z_2)$ $-(z_1 - z_2)^2 = (z_3 - z_1)(z_3 - z_2)$ $-(z_1^2 - 2z_1z_2 + z_2^2) = z_3^2 - z_3z_2 - z_1z_3 + z_1z_2$ $-z_1^2 + 2z_1z_2 - z_2^2 = z_3^2 - z_2z_3 - z_3z_1 + z_1z_2$ $0 = z_1^2 + z_2^2 + z_3^2 - z_1z_2 - z_2z_3 - z_3z_1$. This means $z_1^2 + z_2^2 + z_3^2 = z_1z_2 + z_2z_3 + z_3z_1$.

Alternative Method using $\omega$: If $z_1, z_2, z_3$ are vertices of an equilateral triangle, then either $z_1 + z_2 \omega + z_3 \omega^2 = 0$ or $z_1 + z_2 \omega^2 + z_3 \omega = 0$. (Assuming vertices are taken in anti-clockwise or clockwise order for the second case respectively). If $z_1 + z_2 \omega^2 + z_3 \omega = 0$, then multiply by $z_1$: $z_1^2+z_1 z_2 \omega^2 + z_1 z_3 \omega = 0$. This is not directly the identity.

Let's take the identity $a^2+b^2+c^2-ab-bc-ca=0 \iff a+b\omega+c\omega^2=0$ or $a+b\omega^2+c\omega=0$. Since this identity is symmetrical for $a,b,c$ and $z_1,z_2,z_3$ are interchangeable, the proof of $z_1^2 + z_2^2 + z_3^2 - z_1z_2 - z_2z_3 - z_3z_1 = 0$ is equivalent to showing that $(z_1 + \omega z_2 + \omega^2 z_3)(z_1 + \omega^2 z_2 + \omega z_3) = 0$ or $(z_1-z_2)(z_2-z_3)(z_3-z_1) = 0$ The condition $z_1^2 + z_2^2 + z_3^2 - z_1z_2 - z_2z_3 - z_3z_1 = 0$ is the direct output from the rotation property if we express $(z_1-z_2)^2$ relationship. The previous solution is more direct.

Another relation for equilateral triangles is: $z_1^2+z_2^2+z_3^2 = (z_1+z_2+z_3)^2/3$ ??? No. The identity $(z_1 - z_2)^2 + (z_2 - z_3)^2 + (z_3 - z_1)^2 = 0$ is for when the vertices are $z_1, z_2, z_3$ and the triangle is equilateral. No, that's not general. Consider $(1, \omega, \omega^2)$. $(1-\omega)^2 + (\omega-\omega^2)^2 + (\omega^2-1)^2 = (1-\omega)^2 + \omega^2(1-\omega)^2 + \omega^2(\omega-1)^2 = (1-\omega)^2(1+\omega^2+\omega^2) \ne 0$. It's $(a-b)^2+(b-c)^2+(c-a)^2= -\frac{1}{2}\sum (a-b-c)^2$ No. Ah, the identity is $z_1^2+z_2^2+z_3^2-z_1z_2-z_2z_3-z_3z_1=0$. This is directly equivalent to $ (z_1+z_2\omega+z_3\omega^2)(z_1+z_2\omega^2+z_3\omega)=0$. This form is useful. The statement that $z_1, z_2, z_3$ are vertices of an equilateral triangle implies that for a suitable ordering (say $z_1,z_2,z_3$ anti-clockwise), $z_1-z_2 = (z_3-z_2)e^{i\pi/3}$. Or $z_1 = z_2 + (z_3-z_2)e^{i\pi/3}$. This is equivalent to $\frac{z_1-z_2}{e^{i\pi/3}} = z_3-z_2$. And $z_2 = z_3 + (z_1-z_3)e^{i\pi/3}$. And $z_3 = z_1 + (z_2-z_1)e^{i\pi/3}$. These lead to $z_1^2+z_2^2+z_3^2=z_1z_2+z_2z_3+z_3z_1$. Let's prove it this way: Given an equilateral triangle with vertices $A(z_1)$, $B(z_2)$, $C(z_3)$. Then $AB=BC=CA$. Thus $|z_1-z_2| = |z_2-z_3| = |z_3-z_1|$. Squaring this: $|z_1-z_2|^2 = |z_2-z_3|^2 = |z_3-z_1|^2$. Let $k = e^{i\pi/3}$. Then for $z_1, z_2, z_3$ in cyclic order (say anti-clockwise), $z_1-z_2 = k(z_3-z_2)$ and $z_2-z_3 = k(z_1-z_3)$ and $z_3-z_1 = k(z_2-z_1)$. From $z_1-z_2=k(z_3-z_2)$, we have $z_1 - z_2 - k z_3 + k z_2 = 0 \implies z_1 + (k-1)z_2 - kz_3 = 0$. The easiest derivation: If $z_1, z_2, z_3$ are vertices of an equilateral triangle, then any one vertex can be obtained by rotating another vertex about the third by $\pm \pi/3$. Alternatively, the transformation $\frac{z_3-z_1}{z_2-z_1}=e^{\pm i\pi/3}$. Case 1: $\frac{z_3-z_1}{z_2-z_1} = e^{i\pi/3}$. Let $e^{i\pi/3} = \alpha$. So $z_3-z_1 = \alpha(z_2-z_1)$. And $\frac{z_1-z_2}{z_3-z_2} = e^{i\pi/3}$. So $z_1-z_2 = \alpha(z_3-z_2)$. We have $z_3-z_1 = \alpha z_2 - \alpha z_1$. And $z_1-z_2 = \alpha z_3 - \alpha z_2$. Add the two equations: $(z_3-z_1) + (z_1-z_2) = \alpha z_2 - \alpha z_1 + \alpha z_3 - \alpha z_2$ $z_3-z_2 = \alpha z_3 - \alpha z_1$ $z_3(1-\alpha) = z_2 - \alpha z_1$. This relation is not immediately leading to the required.

The expression $z_1^2 + z_2^2 + z_3^2 - z_1z_2 - z_2z_3 - z_3z_1 = 0$ is what we need to show. We know that $a^2+b^2+c^2-ab-bc-ca = 0 \iff a+b\omega+c\omega^2=0$ or $a+b\omega^2+c\omega=0$. This is equivalent because the LHS is $(a+b\omega+c\omega^2)(a+b\omega^2+c\omega)$. So we need to show that for an equilateral triangle, one of these expressions is zero. If $z_1, z_2, z_3$ form an equilateral triangle in anti-clockwise order, then $z_2-z_1 = (z_3-z_1)e^{i\pi/3}$. This means $z_2-z_1 = z_3 e^{i\pi/3} - z_1 e^{i\pi/3}$. $z_1(e^{i\pi/3}-1) + z_2 - z_3 e^{i\pi/3} = 0$. This is a common method for showing this identity.

Final Attempt at Simplified Proof: Consider the complex numbers $z_1, z_2, z_3$. The condition for an equilateral triangle is that (assuming vertices are in anticlockwise order), rotating $z_3$ about $z_1$ by $\pi/3$ gives $z_2$. So, $(z_2 - z_1) = (z_3 - z_1)e^{i\pi/3}$. Similarly, rotating $z_1$ about $z_2$ by $\pi/3$ gives $z_3$. So, $(z_3 - z_2) = (z_1 - z_2)e^{i\pi/3}$. Let $k = e^{i\pi/3}$. Then $z_2 - z_1 = k(z_3 - z_1)$ (Eq. 1) And $z_3 - z_2 = k(z_1 - z_2)$ (Eq. 2)

From (1): $z_2 - z_1 - k z_3 + k z_1 = 0 \implies (1-k)z_1 + z_2 - k z_3 = 0$. From (2): $z_3 - z_2 - k z_1 + k z_2 = 0 \implies -k z_1 + (k-1)z_2 + z_3 = 0$. These are two linear equations involving $z_1, z_2, z_3$. This algebraic manipulation using rotation forms holds.

Let's use the property that if $z_1, z_2, z_3$ are vertices of an equilateral triangle, then $z_1^2 + z_2^2 + z_3^2 - z_1z_2 - z_2z_3 - z_3z_1 = 0$ This identity is often useful: $a^2+b^2+c^2-ab-bc-ca = \frac{1}{2}[(a-b)^2+(b-c)^2+(c-a)^2]$. So, we need to show $\frac{1}{2}[(z_1-z_2)^2+(z_2-z_3)^2+(z_3-z_1)^2] = 0$. This means $(z_1-z_2)^2+(z_2-z_3)^2+(z_3-z_1)^2 = 0$. Let side length be $L$. Then $L = |z_1-z_2| = |z_2-z_3| = |z_3-z_1|$. The numbers $z_1-z_2$, $z_2-z_3$, $z_3-z_1$ are complex numbers representing the sides. These vectors form a triangle. $z_1-z_2 = A$. $z_2-z_3 = B$. $z_3-z_1 = C$. Note that $A+B+C = 0$. Also $|A|=|B|=|C|=L$. If $A, B, C$ are vectors forming an equilateral triangle, then the angles between them are $120^\circ$. So $B = A e^{i2\pi/3} = A\omega$ and $C = B e^{i2\pi/3} = B\omega = A\omega^2$. Then $A^2 + B^2 + C^2 = A^2 + (A\omega)^2 + (A\omega^2)^2 = A^2(1 + \omega^2 + \omega^4)$. Since $\omega^3=1$, $\omega^4=\omega$. So $A^2(1 + \omega^2 + \omega) = A^2(0) = 0$. Therefore, $(z_1-z_2)^2+(z_2-z_3)^2+(z_3-z_1)^2 = 0$. This is probably the most elegant way to prove it.

Example 5 (Sequences and Series - AGP Sum):

Find the sum of the series $S_\infty = 1 + \frac{2}{2!} + \frac{3}{3!} + \frac{4}{4!} + \dots$

Solution: The general term of the series is $T_n = \frac{n}{n!}$. We can simplify $T_n$: $T_n = \frac{n}{n \cdot (n-1)!} = \frac{1}{(n-1)!}$ for $n \ge 1$. For $n = 1$, $T_1 = \frac{1}{0!} = \frac{1}{1} = 1$. For $n = 2$, $T_2 = \frac{2}{2!} = \frac{1}{1!} = 1$. For $n = 3$, $T_3 = \frac{3}{3!} = \frac{1}{2!} = \frac{1}{2}$. The series becomes $S_\infty = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \dots$. This is the well-known Maclaurin series for $e^x$ evaluated at $x=1$. $e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = \frac{x^0}{0!} + \frac{x^1}{1!} + \frac{x^2}{2!} + \dots$ For $x=1$, $e^1 = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \dots$. So, $S_\infty = e$.

Answer: The sum of the series is $e$.

Example 6 (Matrices & Determinants - Properties):

Let $A$ be a $3 \times 3$ matrix with real entries. If $A^2 = A$ and $\det(A) \ne 0$, then show that $A=I$.

Solution: Given $A^2 = A$. This means $A$ is an idempotent matrix. Also given $\det(A) \ne 0$. This implies that $A$ is an invertible matrix. Since $A$ is invertible, $A^{-1}$ exists. Multiply the equation $A^2=A$ by $A^{-1}$ from the left (or right): $A^{-1}(A^2) = A^{-1}A$ $A^{-1}(A \cdot A) = I$ $(A^{-1}A)A = I$ $IA = I$ $A = I$.

This shows that the only non-singular idempotent matrix is the identity matrix.

Answer: If $A^2=A$ and $\det(A) \ne 0$, then $A=I$.

Common Mistakes/Pitfalls

• Complex Numbers:
  • Treating $i$ as a variable in algebraic factorization without considering $i^2=-1$.
  • Incorrectly identifying principal argument (sign and quadrant errors).
  • Errors in converting between Cartesian and Polar forms.
  • Assuming $|z_1+z_2| = |z_1|+|z_2|$ always (only if collinear and in same direction).
• Sequences & Series:
  • Confusing formula for $n$-th term versus sum of $n$ terms.
  • Forgetting conditions for infinite GP sum ($|r|<1$).
  • Algebraic errors in solving equations for $a, d, r$.
  • Not noting restrictions like positive terms.
• Matrices & Determinants:
  • Assuming matrix multiplication is commutative.
  • Errors in calculating determinant or adjoint for larger matrices.
  • Forgetting $\det(kA) = k^n \det(A)$.
  • Incorrectly reasoning about consistency of linear equations when $\det(A)=0$.
  • Mistakes with transpose properties, especially $(AB)^T = B^TA^T$.

Memory Tricks and Mnemonics

• De Moivre's Theorem: "De Moivre's makes the 'n' move inside 'theta'": For $(\cos\theta + i\sin\theta)^n$, the $n$ goes inside to make $\cos(n\theta) + i\sin(n\theta)$.
• AM-GM-HM inequality: Alphabetical order: Arithmetic $\ge$ Geometric $\ge$ Harmonic.
• Determinant properties:
  • 'Row operations don't change value, column ops don't change value, just scale for multiplication.'
  • 'Identical rows/columns mean zero determinant.'
• Adjoint/Inverse formulas:
  • $A \cdot (\text{adj} A) = (\det A) I$ (relates the matrix to its adjoint and inverse).
  • Inverse formula: "$1$ over $\det(A)$, times $\text{adj}(A)$".
• Equilateral Triangle Condition (Complex Numbers): $z_1^2 + z_2^2 + z_3^2 = z_1z_2 + z_2z_3 + z_3z_1$ is a key identity.
• Sum of AP terms: Average of first and last term, times number of terms: $\frac{a_1+a_n}{2} \cdot n$.

Previous Year Question Patterns

• Complex Numbers:
  • Locus problems involving inequalities ($|z-z_0| < r$) or equations ($|z-z_1| = |z-z_2|$).
  • Properties of modulus and argument, often combined with algebraic manipulation.
  • Cube roots of unity, $n$-th roots of unity properties.
  • De Moivre's Theorem applications, sometimes for trigonometric identities.
• Sequences & Series:
  • Sum to infinity of GP or AGP.
  • Problems involving AM-GM-HM inequality to find maximum/minimum values.
  • Finding the $n$-th term or sum of series using the method of differences or by recognizing standard series.
  • Word problems often translated into AP/GP scenarios.
• Matrices & Determinants:
  • System of linear equations (Cramer's rule, matrix inverse method, consistency conditions).
  • Properties of adjoint and inverse (e.g., $\det(\text{adj}A)$, $(\text{adj}A)^{-1}$, etc.).
  • Properties of determinants for simplification and evaluation (esp. $3 \times 3$ or $4 \times 4$ matrices).
  • Matrix equations and operations (e.g., $A^2=A$, $(A+B)^2$).
  • Special types of matrices (symmetric, skew-symmetric, orthogonal, idempotent, etc.).

Revision Checklist

Complex Numbers

1. Definitions: $z = x+iy$, $\bar{z}$, $|z|$, $\text{arg}(z)$ (principal).
2. Forms: Cartesian, Polar, Euler.
3. Operations: Addition, subtraction, multiplication, division, powers.
4. Properties: Modulus properties ($|z_1z_2|$, triangle inequality), Argument properties.
5. De Moivre's Theorem: $(e^{i\theta})^n = e^{in\theta}$.
6. Roots of Unity: $n$-th roots, cube roots ($\omega, \omega^2$) and their properties.
7. Geometry: Loci of complex numbers, rotation, distance.
8. Square Roots of $z$.

Sequences & Series

1. AP: $a_n$, $S_n$, properties, AM.
2. GP: $a_n$, $S_n$, $S_\infty$, properties, GM.
3. HP: Definition, HM, relation to AP.
4. AM-GM-HM: Inequality and conditions for equality.
5. AGP: General form, sum to $n$ terms, sum to infinity.
6. Special Series: $\sum n, \sum n^2, \sum n^3$.
7. Method of Differences.

Matrices & Determinants

1. Matrix Types: Row, Column, Square, Identity, Zero, Diagonal, Scalar, Upper/Lower Triangular, Symmetric, Skew-symmetric, Orthogonal, Idempotent, Nilpotent, Involutory.
2. Operations: Addition/Subtraction, Scalar Multiplication, Matrix Multiplication, Transpose.
3. Determinants: Calculation for $2 \times 2, 3 \times 3$, minors, cofactors.
4. Properties of Determinants: Effect of row/column ops, $\det(A^T)$, $\det(kA)$, $\det(AB)$.
5. Adjoint: Definition, properties related to $\det(A)$ and $A$.
6. Inverse: Formula, existence condition ($\det(A) \ne 0$), properties.
7. System of Linear Equations: Cramer's Rule, Matrix Method ($AX=B$), consistency conditions (unique, infinite, no solution), homogeneous systems.

Last-Day Strategy

1. Quick Scan: Go through this cheat sheet/your own notes to refresh all formulas and key concepts. Don't try to learn new topics.
2. Targeted Practice (Mental): Instead of solving full problems, mentally trace the steps for 2-3 problems for each sub-topic. Focus on identifying the correct formula/concept application.
3. Review Common Mistakes: Briefly look over the 'Common Mistakes' section to avoid silly errors under pressure.
4. Important Identities: Ensure important identities for complex numbers ($\omega$ relations, equilateral triangle), series sums, and matrix determinant properties are at your fingertips.
5. Stay Calm: Trust your preparation. A good night's sleep is more valuable than cramming new material. Visualise yourself solving the problems correctly.
6. Time Management: In the exam, read questions carefully. Prioritize easy-to-medium problems. Don't get stuck on one problem for too long. For multi-concept problems, break them down.

Good luck! You've got this.