Matrices

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Types of Matrices:

• Row/Column: $1 \times n$ or $n \times 1$ matrix
• Square: $n \times n$ (same rows and columns)
• Zero matrix: All entries are 0
• Identity: Diagonal entries 1, others 0; denoted $I_n$
• Diagonal: Non-diagonal entries are 0
• Symmetric: $A^T = A$
• Skew-symmetric: $A^T = -A$

Operations:

• Addition: Same-size matrices, element-wise
• Scalar multiplication: Multiply each element by scalar
• Multiplication: $C_{ij} = \sum_k A_{ik} B_{kj}$; number of columns of $A$ must equal number of rows of $B$
• Transpose: Swap rows and columns

Determinant (for $2 \times 2$): $$|A| = \begin{vmatrix} a & b \ c & d \end{vmatrix} = ad - bc$$

⚡ JEE Tip: Matrix multiplication is NOT commutative: $AB \neq BA$ in general. Always check order when multiplying matrices.

⚡ Common Mistake: $|A+B| \neq |A| + |B|$. Determinant is multiplicative: $|AB| = |A||B|$, but not additive.

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding.

Properties of Determinants:

1. $|A^T| = |A|$
2. $|AB| = |A||B|$
3. $|A^{-1}| = 1/|A|$ (if $A$ is invertible)
4. Swapping two rows changes sign
5. Multiplying a row by scalar $k$: $|kA| = k^n|A|$ for $n \times n$
6. Adding multiple of one row to another: determinant unchanged

Adjoint and Inverse:

For a square matrix $A$:

• Adjoint: $\text{adj}(A) = C^T$ where $C_{ij} = (-1)^{i+j}M_{ij}$ (cofactor matrix, transposed)
• Inverse: $A^{-1} = \frac{\text{adj}(A)}{|A|}$ (provided $|A| \neq 0$)

$A$ is invertible (non-singular) iff $|A| \neq 0$.

Rank of Matrix:

The rank $r(A)$ is the number of non-zero rows in its row echelon form.

• Maximum rank is $\min(m,n)$ for $m \times n$ matrix
• $r(A) = r$ means there exists at least one $r \times r$ minor with non-zero determinant, and all $(r+1) \times (r+1)$ minors are zero

Worked Examples:

Example 1: If $A = \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}$, find $A^{-1}$.

$|A| = 1 \cdot 4 - 2 \cdot 3 = 4 - 6 = -2$. $\text{adj}(A) = \begin{pmatrix} 4 & -2 \ -3 & 1 \end{pmatrix}^T = \begin{pmatrix} 4 & -3 \ -2 & 1 \end{pmatrix}$? Wait, cofactor matrix: $C_{11} = 4, C_{12} = -3, C_{21} = -2, C_{22} = 1$. So cofactor matrix $= \begin{pmatrix} 4 & -3 \ -2 & 1 \end{pmatrix}$. $\text{adj}(A) = \begin{pmatrix} 4 & -2 \ -3 & 1 \end{pmatrix}$… wait the off-diagonal need negation.

Actually: $C_{11} = 4, C_{12} = -3$ (since $(-1)^{1+2} \cdot 3$), $C_{21} = -2$ (since $(-1)^{2+1} \cdot 2$), $C_{22} = 1$. So $C = \begin{pmatrix} 4 & -3 \ -2 & 1 \end{pmatrix}$. $\text{adj}(A) = C^T = \begin{pmatrix} 4 & -2 \ -3 & 1 \end{pmatrix}$.

$A^{-1} = \frac{1}{-2} \begin{pmatrix} 4 & -2 \ -3 & 1 \end{pmatrix} = \begin{pmatrix} -2 & 1 \ 3/2 & -1/2 \end{pmatrix}$.

Example 2 (JEE 2021): Find rank of $A = \begin{pmatrix} 1 & 2 & 3 \ 2 & 4 & 6 \ 3 & 6 & 9 \end{pmatrix}$.

Note that row 2 = 2(row 1) and row 3 = 3(row 1). So rank is at most 1. Check: is there a non-zero element? Yes, $A_{11} = 1 \neq 0$. So rank = 1.

Example 3: Solve system using matrix method: $2x + y = 5$ $x + 2y = 4$

Matrix form: $\begin{pmatrix} 2 & 1 \ 1 & 2 \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} 5 \ 4 \end{pmatrix}$. $|A| = 4 - 1 = 3 \neq 0$. $A^{-1} = \frac{1}{3}\begin{pmatrix} 2 & -1 \ -1 & 2 \end{pmatrix}$. $\begin{pmatrix} x \ y \end{pmatrix} = A^{-1} \begin{pmatrix} 5 \ 4 \end{pmatrix} = \frac{1}{3}\begin{pmatrix} 2 & -1 \ -1 & 2 \end{pmatrix}\begin{pmatrix} 5 \ 4 \end{pmatrix} = \frac{1}{3}\begin{pmatrix} 10 - 4 \ -5 + 8 \end{pmatrix} = \frac{1}{3}\begin{pmatrix} 6 \ 3 \end{pmatrix} = \begin{pmatrix} 2 \ 1 \end{pmatrix}$.

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🔴 Extended — Deep Study (3mo+)

Comprehensive theory for serious JEE Advanced preparation.

Eigenvalues and Eigenvectors:

For square matrix $A$, if $A\mathbf{v} = \lambda \mathbf{v}$ for non-zero vector $\mathbf{v}$, then $\lambda$ is an eigenvalue and $\mathbf{v}$ is an eigenvector.

Characteristic Equation: $$|A - \lambda I| = 0$$

This gives a polynomial of degree $n$ in $\lambda$. Sum of eigenvalues (trace) $= \text{tr}(A) = \sum A_{ii}$. Product of eigenvalues $= |A|$.

Properties:

• Sum of eigenvalues = trace
• Product of eigenvalues = determinant
• For symmetric matrix, all eigenvalues are real
• For skew-symmetric matrix, eigenvalues are purely imaginary or zero

Cayley-Hamilton Theorem: Every square matrix satisfies its own characteristic equation: $p(\lambda) = |\lambda I - A| = \lambda^n + c_1\lambda^{n-1} + \cdots + c_n$. Then $p(A) = A^n + c_1 A^{n-1} + \cdots + c_n I = 0$.

This allows computing powers of matrices efficiently.

Diagonalisation:

$A = PDP^{-1}$ where $D$ is diagonal with eigenvalues. $A^n = PD^nP^{-1}$.

Worked Example:

JEE Advanced 2019: If $A = \begin{pmatrix} 2 & 1 \ 1 & 2 \end{pmatrix}$, find $A^n$.

Find eigenvalues: $|A - \lambda I| = \begin{vmatrix} 2-\lambda & 1 \ 1 & 2-\lambda \end{vmatrix} = (2-\lambda)^2 - 1 = \lambda^2 - 4\lambda + 3 = (\lambda-1)(\lambda-3) = 0$. So $\lambda_1 = 1, \lambda_2 = 3$.

For $\lambda_1 = 1$: $(A-I)\mathbf{v} = 0$ → $\begin{pmatrix} 1 & 1 \ 1 & 1 \end{pmatrix}\mathbf{v} = 0$ → $v_1 + v_2 = 0$. Eigenvector $\mathbf{v}_1 = \begin{pmatrix} 1 \ -1 \end{pmatrix}$.

For $\lambda_2 = 3$: $(A-3I)\mathbf{v} = 0$ → $\begin{pmatrix} -1 & 1 \ 1 & -1 \end{pmatrix}\mathbf{v} = 0$ → $-v_1 + v_2 = 0$. Eigenvector $\mathbf{v}_2 = \begin{pmatrix} 1 \ 1 \end{pmatrix}$.

$P = \begin{pmatrix} 1 & 1 \ -1 & 1 \end{pmatrix}$, $D = \begin{pmatrix} 1 & 0 \ 0 & 3 \end{pmatrix}$. $P^{-1} = \frac{1}{2}\begin{pmatrix} 1 & -1 \ 1 & 1 \end{pmatrix}$.

$A^n = P D^n P^{-1} = \frac{1}{2}\begin{pmatrix} 1 & 1 \ -1 & 1 \end{pmatrix}\begin{pmatrix} 1 & 0 \ 0 & 3^n \end{pmatrix}\begin{pmatrix} 1 & -1 \ 1 & 1 \end{pmatrix}$. $= \frac{1}{2}\begin{pmatrix} 1 & 3^n \ -1 & 3^n \end{pmatrix}\begin{pmatrix} 1 & -1 \ 1 & 1 \end{pmatrix} = \frac{1}{2}\begin{pmatrix} 1+3^n & -1+3^n \ -1+3^n & 1+3^n \end{pmatrix}$.

So $A^n = \frac{1}{2}\begin{pmatrix} 3^n+1 & 3^n-1 \ 3^n-1 & 3^n+1 \end{pmatrix}$.

JEE Advanced Patterns (2018–2024):

• Solving systems using matrices and Cramer’s rule is common
• Eigenvalue problems appeared in 2019, 2021, 2023
• Cayley-Hamilton theorem is frequently tested (to compute $A^n$)
• Rank and consistency of linear equations are common
• Orthogonal and unitary matrices appear in advanced sets

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