Matrices and Determinants are fundamental topics in linear algebra and form a crucial part of the JEE Main syllabus. A strong understanding of these concepts, including their properties, operations, and applications, is essential for solving a wide variety of problems. This formula sheet summarizes the key definitions, theorems, and results pertinent to cracking the JEE Main examination.
Key Concepts
Matrix: A rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns. Order of a Matrix: The number of rows ($m$) by the number of columns ($n$), denoted as $m \times n$. Types of Matrices:
- Row Matrix: A matrix with only one row ($1 \times n$).
- Column Matrix: A matrix with only one column ($m \times 1$).
- Square Matrix: A matrix where the number of rows equals the number of columns ($m=n$).
- Zero/Null Matrix: A matrix where all elements are zero.
- Diagonal Matrix: A square matrix where all non-diagonal elements are zero.
- Scalar Matrix: A diagonal matrix where all diagonal elements are equal.
- Identity/Unit Matrix: A scalar matrix where all diagonal elements are 1, denoted by $I_n$.
- Upper Triangular Matrix: A square matrix where all elements below the main diagonal are zero ($a_{ij}=0$ for $i>j$).
- Lower Triangular Matrix: A square matrix where all elements above the main diagonal are zero ($a_{ij}=0$ for $i<j$).
- Symmetric Matrix: A square matrix where $A^T = A$, meaning $a_{ij} = a_{ji}$.
- Skew-Symmetric Matrix: A square matrix where $A^T = -A$, meaning $a_{ij} = -a_{ji}$. Diagonal elements of a skew-symmetric matrix are always zero.
- Orthogonal Matrix: A square matrix such that $AA^T = A^T A = I$. Implies $A^{-1} = A^T$.
- Idempotent Matrix: A square matrix such that $A^2 = A$.
- Nilpotent Matrix: A square matrix such that $A^k = O$ for some positive integer $k$. The least such $k$ is the index of nilpotency.
- Involutory Matrix: A square matrix such that $A^2 = I$.
Determinant: A scalar value that can be computed from the elements of a square matrix. It provides important information about the matrix, such as its invertibility. Singular Matrix: A square matrix whose determinant is zero ($|A|=0$). It does not have an inverse. Non-Singular Matrix: A square matrix whose determinant is non-zero ($|A| \neq 0$). It has an inverse. Minors and Cofactors:
- Minor ($M_{ij}$): The determinant of the submatrix obtained by deleting the $i$-th row and $j$-th column.
- Cofactor ($C_{ij}$): $C_{ij} = (-1)^{i+j} M_{ij}$. Adjoint of a Matrix: The transpose of the cofactor matrix. Inverse of a Matrix: For a non-singular square matrix $A$, its inverse $A^{-1}$ satisfies $AA^{-1} = A^{-1}A = I$. System of Linear Equations: Matrices and determinants are used to solve systems of linear equations using methods like Cramer's Rule or Matrix Inverse Method. Homogeneous System: A system of linear equations where all constant terms are zero.
Formulas & Results
Matrix Addition/Subtraction: For two matrices $A = [a_{ij}]{m \times n}$ and $B = [b{ij}]{m \times n}$, $A \pm B = [a{ij} \pm b_{ij}]_{m \times n}$
Scalar Multiplication: For a matrix $A = [a_{ij}]{m \times n}$ and a scalar $k$, $kA = [ka{ij}]_{m \times n}$
Matrix Multiplication: For $A = [a_{ij}]{m \times p}$ and $B = [b{ij}]{p \times n}$, $C=AB = [c{ij}]{m \times n}$ where $c{ij} = \sum_{k=1}^{p} a_{ik}b_{kj}$ (Number of columns of A must equal number of rows of B)
Properties of Transpose:
- $(A^T)^T = A$
- $(A \pm B)^T = A^T \pm B^T$
- $(kA)^T = kA^T$
- $(AB)^T = B^T A^T$
Determinant of a $2 \times 2$ Matrix: For $A = \begin{pmatrix} a & b \ c & d \end{pmatrix}$, $|A| = ad - bc$
Determinant of a $3 \times 3$ Matrix: For $A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{pmatrix}$, $|A| = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})$
Properties of Determinants:
- $|A^T| = |A|$
- If any two rows (or columns) are interchanged, the sign of the determinant changes.
- If any two rows (or columns) are identical, $|A| = 0$.
- If all elements of a row (or column) are zero, $|A| = 0$.
- If elements of a row (or column) are multiplied by a scalar $k$, the determinant is multiplied by $k$.
- $|kA| = k^n |A|$ where $n$ is the order of the matrix.
- $|AB| = |A||B|$
- If $A$ is a triangular matrix (upper or lower), $|A|$ is the product of its diagonal elements.
- Determinant remains unchanged if any multiple of one row (or column) is added to another row (or column).
Adjoint of a Matrix: If $A = [a_{ij}]$ and $C = [C_{ij}]$ is its cofactor matrix, then $\text{adj}(A) = C^T$
Relation between Matrix, Adjoint, and Inverse: $A , \text{adj}(A) = \text{adj}(A) , A = |A|I$
Inverse of a Matrix: For a non-singular matrix $A$, $A^{-1} = \frac{1}{|A|} \text{adj}(A)$
Properties of Inverse:
- $(A^{-1})^{-1} = A$
- $(AB)^{-1} = B^{-1}A^{-1}$
- $(kA)^{-1} = \frac{1}{k} A^{-1}$
- $(A^T)^{-1} = (A^{-1})^T$
- $|A^{-1}| = \frac{1}{|A|}$
For an orthogonal matrix $A$ (i.e. $AA^T = I$): $A^{-1} = A^T$
For an Idempotent Matrix $A$ (i.e. $A^2=A$): If $A$ is non-singular, then $A=I$. (Since $A^2=A \implies A^{-1}A^2=A^{-1}A \implies A=I$)
System of Linear Equations (Matrix Method): For $AX = B$:
- If $|A| \neq 0$, a unique solution exists: $X = A^{-1}B$. (Consistent)
- If $|A| = 0$:
- If $(\text{adj}(A))B \neq O$, no solution exists. (Inconsistent)
- If $(\text{adj}(A))B = O$, infinitely many solutions exist. (Consistent and dependent)
System of Linear Equations (Cramer's Rule for $AX = B$): Let $D = |A|$. Let $D_j$ be the determinant obtained by replacing the $j$-th column of $A$ with $B$.
- If $D \neq 0$, unique solution: $x_j = D_j/D$.
- If $D = 0$ and at least one $D_j \neq 0$, no solution.
- If $D = 0$ and all $D_j = 0$, infinitely many solutions or no solution (further analysis needed by checking consistency).
Homogeneous System $AX = O$:
- If $|A| \neq 0$, only trivial solution ($X = O$).
- If $|A| = 0$, non-trivial solutions exist (infinitely many).
Worked Examples
Example 1: If $A = \begin{pmatrix} 2 & -3 \ 1 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 0 \ 2 & 5 \end{pmatrix}$, find $AB$ and $|AB|$.
Solution: First, calculate $AB$: $AB = \begin{pmatrix} 2 & -3 \ 1 & 4 \end{pmatrix} \begin{pmatrix} 1 & 0 \ 2 & 5 \end{pmatrix} = \begin{pmatrix} (2)(1) + (-3)(2) & (2)(0) + (-3)(5) \ (1)(1) + (4)(2) & (1)(0) + (4)(5) \end{pmatrix}$ $AB = \begin{pmatrix} 2 - 6 & 0 - 15 \ 1 + 8 & 0 + 20 \end{pmatrix} = \begin{pmatrix} -4 & -15 \ 9 & 20 \end{pmatrix}$
Now, calculate $|AB|$: $|AB| = (-4)(20) - (-15)(9) = -80 - (-135) = -80 + 135 = 55$.
Alternatively, using the property $|AB| = |A||B|$: $|A| = (2)(4) - (-3)(1) = 8 - (-3) = 11$ $|B| = (1)(5) - (0)(2) = 5 - 0 = 5$ $|AB| = |A||B| = (11)(5) = 55$.
Example 2: Find the value of $x$ for which the matrix $A = \begin{pmatrix} 1 & -2 & 3 \ 1 & 2 & 1 \ x & 2 & -3 \end{pmatrix}$ is singular.
Solution: A matrix is singular if its determinant is zero. $|A| = 1((2)(-3) - (1)(2)) - (-2)((1)(-3) - (1)(x)) + 3((1)(2) - (2)(x))$ $|A| = 1(-6 - 2) + 2(-3 - x) + 3(2 - 2x)$ $|A| = 1(-8) + 2(-3 - x) + 3(2 - 2x)$ $|A| = -8 - 6 - 2x + 6 - 6x$ $|A| = -8 - 8x$
For $A$ to be singular, $|A|=0$: $-8 - 8x = 0$ $-8x = 8$ $x = -1$.
Example 3: If $A = \begin{pmatrix} 3 & 1 \ 7 & 5 \end{pmatrix}$, find $A^{-1}$.
Solution: First, calculate the determinant of $A$: $|A| = (3)(5) - (1)(7) = 15 - 7 = 8$. Since $|A| \neq 0$, the inverse exists.
Next, find the adjoint of $A$. For a $2 \times 2$ matrix $A = \begin{pmatrix} a & b \ c & d \end{pmatrix}$, $\text{adj}(A) = \begin{pmatrix} d & -b \ -c & a \end{pmatrix}$. So, $\text{adj}(A) = \begin{pmatrix} 5 & -1 \ -7 & 3 \end{pmatrix}$.
Now, use the formula $A^{-1} = \frac{1}{|A|} \text{adj}(A)$: $A^{-1} = \frac{1}{8} \begin{pmatrix} 5 & -1 \ -7 & 3 \end{pmatrix} = \begin{pmatrix} 5/8 & -1/8 \ -7/8 & 3/8 \end{pmatrix}$.
Common Mistakes
- Order of Matrix Multiplication: For $AB$, the number of columns of $A$ must equal the number of rows of $B$. $AB \neq BA$ in general.
- Sign Errors in Cofactors: For $C_{ij} = (-1)^{i+j} M_{ij}$, incorrectly determining the sign is a frequent error.
- Calculating Determinants: Especially for $3 \times 3$ matrices, care must be taken with signs and terms. Sarrus' Rule can be helpful but also prone to error if not applied correctly.
- Formula for $A^{-1}$: Forgetting to divide by $|A|$ or using the wrong formula for $\text{adj}(A)$ (e.g., not transposing the cofactor matrix).
- Properties of Determinants: Incorrectly applying linearity (e.g., $|A+B| \neq |A|+|B|$) or scalar multiplication ($|kA| = k^n|A|$, not $k|A|$).
- Singular Matrix Condition: Confusing non-zero determinant for singular matrix or vice-versa. Remember, $|A|=0$ means singular, no inverse.
- Cramer's Rule Conditions: Not understanding when unique solutions, no solutions, or infinitely many solutions exist based on the values of $D$ and $D_j$.
Quick Revision Checklist
- Understand different types of matrices and their definitions (square, diagonal, identity, symmetric, skew-symmetric, orthogonal, idempotent, nilpotent, involutory).
- Master matrix arithmetic: addition, subtraction, scalar multiplication, and matrix multiplication.
- Know properties of transpose, especially $(AB)^T = B^T A^T$.
- Be proficient in calculating determinants of $2 \times 2$ and $3 \times 3$ matrices.
- Memorize key determinant properties, especially $|kA|=k^n|A|$ and $|AB|=|A||B|$.
- Accurately find minors and cofactors for all elements.
- Correctly calculate the adjoint of a matrix.
- Understand the formula for $A^{-1} = \frac{1}{|A|} \text{adj}(A)$ and when it exists.
- Recall properties of inverse, particularly $(AB)^{-1} = B^{-1}A^{-1}$.
- Know the conditions for consistent/inconsistent and unique/infinite solutions for systems of linear equations using both matrix inverse method and Cramer's Rule.
- Understand conditions for trivial and non-trivial solutions in homogeneous systems.
