Class 12 Mathematics Chapter 3: Matrices

CBSE 2026–27 | NCERT-aligned free resource

1. Matrix Basics

A matrix is a rectangular arrangement of numbers in rows and columns. Its order is m×n when it has m rows and n columns.

2. Types

Important types include row, column, rectangular, square, zero, diagonal, scalar, identity, symmetric and skew-symmetric matrices.

3. Operations

Addition requires equal order. Matrix multiplication is possible when the number of columns of the first matrix equals the number of rows of the second. In general AB≠BA.

4. Transpose

The transpose Aᵀ is formed by interchanging rows and columns. A symmetric matrix satisfies Aᵀ=A; a skew-symmetric matrix satisfies Aᵀ=−A.

5. Inverse

A square matrix A is invertible if there exists A⁻¹ such that AA⁻¹=A⁻¹A=I. For an invertible matrix, the inverse is unique.

Worked Example

For A=[[1,2],[3,4]], multiplication with another compatible matrix is carried out row-by-column. The order of multiplication matters; reversing the matrices can produce a different result.

Common Mistakes

• Do not multiply matrices element-by-element.
• Check dimensions before multiplication.
• AB=BA is not generally true.

Practice

1. Identify the order and type of a matrix.
2. Find transpose and classify symmetric/skew-symmetric matrices.
3. Multiply compatible matrices.
4. Find the inverse of a nonsingular 2×2 matrix.

Deep-Dive Revision & Worked Examples

Matrix Multiplication

If A is m×n and B is n×p, then AB exists and is m×p. The inner dimensions must match. Matrix multiplication is generally not commutative.

Worked Example

Let A=[[1,2],[3,4]] and B=[[2,0],[1,5]]. Then AB=[[4,10],[10,20]], while BA=[[2,4],[16,22]]. Thus AB≠BA.

Inverse of a 2×2 Matrix

For A=[[a,b],[c,d]], if ad−bc≠0, A⁻¹=1/(ad−bc)[[d,−b],[−c,a]].

Visual Structure

Applications

Matrix equations can represent systems of linear equations. Writing AX=B allows systematic solution when A is invertible: X=A⁻¹B.

Practice

1. Check whether two matrices can be multiplied.
2. Calculate AB and BA and compare them.
3. Find the inverse of a 2×2 matrix.
4. Solve a system using matrix notation.
5. Verify a matrix identity by direct multiplication.