Summary & JEE Exam Tips — Class 12 Math Notes

Matrix: A rectangular array of numbers or functions. A matrix with 'm' rows and 'n' columns has an order of m x n.
Types of Matrices: Key types include Row, Column, Square, Diagonal, Scalar, Identity (I), and Zero (O) matrices.
Matrix Operations:
- Addition/Subtraction: Possible only for matrices of the same order. Operations are performed element-wise. Addition is commutative and associative.
- Scalar Multiplication: Multiply every element by the scalar.
- Matrix Multiplication: The product AB is defined if columns of A = rows of B. If A is m x n and B is n x p, then AB is m x p. Multiplication is associative but not commutative in general ( $AB \ne BA$ ).
Transpose ( $A^T$ ): The matrix obtained by interchanging rows and columns. Key properties include $(A^T)^T=A$ and the reversal law $(AB)^T = B^T A^T$ .
Symmetric & Skew-Symmetric Matrices:
- Symmetric: A square matrix where $A^T = A$ (i.e., $a_{ij} = a_{ji}$ ).
- Skew-Symmetric: A square matrix where $A^T = -A$ (i.e., $a_{ij} = -a_{ji}$ ). Diagonal elements are always zero.
- Any square matrix can be expressed as the sum of a symmetric and a skew-symmetric matrix: $A = \frac{1}{2}(A+A^T) + \frac{1}{2}(A-A^T)$ .
Invertible Matrix: A square matrix A is invertible if there exists a matrix $A^{-1}$ such that $AA^{-1} = A^{-1}A = I$ . A matrix has an inverse if and only if its determinant is non-zero (covered in the next chapter).

Matrix Multiplication is Not Commutative: This is the single most important property to remember. Many mistakes and tricky questions are based on the fact that $AB \ne BA$ . Always be careful with the order of multiplication.
Check Dimensions Before Multiplying: Before you start a matrix multiplication, quickly check the orders of the matrices to ensure the product is defined and to know the order of the resulting matrix. This can save you from lengthy incorrect calculations.
Use Properties of Transpose and Inverse: Problems are often simplified by using properties like $(AB)^T = B^TA^T$ and $(AB)^{-1} = B^{-1}A^{-1}$ instead of calculating the products directly.
Master Symmetric and Skew-Symmetric Properties: Questions involving the decomposition of a matrix into symmetric and skew-symmetric parts are common. Also, remember that the diagonal of a skew-symmetric matrix is always all zeros.
Matrix Polynomials: For questions involving powers of a matrix (like finding $A^{50}$ ), don't start multiplying repeatedly. Instead, calculate $A^2$ , $A^3$ and look for a pattern or a repeating cycle. Often, you will find that $A^k$ simplifies to something like $nA$ or a relation like $A^2 - A + I = O$ , which can be used to find higher powers.