Class 12 Maths Notes

Matrices Class 12 Notes

Complete, exam-ready notes on matrices: notation and types of matrices, addition and scalar multiplication, the row-by-column rule for multiplication, transpose and symmetric/skew-symmetric matrices, and the adjoint-inverse formula — written for CBSE boards and JEE revision.

Class12SubjectMathematicsCoversCBSE · JEE

Written byDeep Narayan· Science & Mathematics EducatorReviewed byPushpanjali

What is this chapter about in one line?

Matrices are rectangular arrays of numbers with their own algebra — addition, multiplication, transposes and inverses — used to solve systems of linear equations efficiently.

Notation and Types of Matrices

Matrix

A rectangular array arranged into m rows and n columns, written $A = [a_{ij}]_{m\times n}$ , where $a_{ij}$ is the entry in row i, column j. The order is m × n.

Row and column matrices have one row or one column; a square matrix has m = n.
Diagonal matrix: all off-diagonal entries zero; identity I: diagonal entries 1.
Scalar and zero matrices: all diagonal entries equal (scalar) or every entry zero.
A symmetric matrix satisfies Aᵀ = A; a skew-symmetric matrix satisfies Aᵀ = −A. The main diagonal of a skew-symmetric matrix is always zero.

A = [a_{ij}]_{m\times n},\quad I_n = \text{diag}(1,1,\dots,1)

Matrix notation and identity

Addition and Scalar Multiplication

Two matrices add only if they have the same order:
$(A+B)_{ij} = a_{ij} + b_{ij}$
.
Scalar multiplication:
$(kA)_{ij} = k\,a_{ij}$
.
Addition is commutative and associative; it distributes over scalar multiplication.
The transpose of a sum:
$(A+B)^T = A^T + B^T$
.

(A+B)_{ij} = a_{ij} + b_{ij},\quad (kA)_{ij} = k\,a_{ij}

Addition and scalar multiplication

Multiplication of Matrices

Multiplication rule

AB is defined only when the number of columns of A equals the number of rows of B: $(m \times n)(n \times p) = (m \times p)$ . Each entry is $(AB)_{ij} = \sum_k a_{ik}b_{kj}$ .

Matrix multiplication is NOT commutative: AB ≠ BA in general.
It is associative and distributive over addition.
A is orthogonal to being invertible separately:
$(AB)^{-1} = B^{-1}A^{-1}$
.
Transpose of a product:
$(AB)^T = B^T A^T$
.

Dimension compatibility

Always check the inner dimensions match: for AB, columns of A must equal rows of B. Watch the reversal in (AB)ᵀ = BᵀAᵀ and (AB)⁻¹ = B⁻¹A⁻¹.

Transpose, Symmetric and Invertibility

Adjoint and inverse

A^{-1}

The inverse of a square matrix exists when its determinant is non-zero: $A^{-1} = \frac{\text{adj}(A)}{|A|}$ . The adjoint adj(A) is the transpose of the matrix of cofactors.

A square matrix A is invertible (non-singular) iff
$|A| \neq 0$
.
Every square matrix is the sum of a symmetric and a skew-symmetric matrix:
$A = \frac{A+A^T}{2} + \frac{A-A^T}{2}$
.
For an invertible matrix,
$A A^{-1} = A^{-1}A = I$
.

A^{-1} = \frac{\text{adj}(A)}{|A|},\quad A = \frac{A+A^T}{2} + \frac{A-A^T}{2}

Inverse and symmetric/skew decomposition

Solved Examples

Example: For $A = \begin{pmatrix}2 & 3 \\ 1 & 4\end{pmatrix}$ , find |A| and A⁻¹.

Solution: $|A| = 2\cdot4 - 3\cdot1 = 5 \neq 0$ , so A is invertible. $\text{adj}(A) = \begin{pmatrix}4 & -3 \\ -1 & 2\end{pmatrix}$ , and $A^{-1} = \frac{1}{5}\begin{pmatrix}4 & -3 \\ -1 & 2\end{pmatrix}$ .

Revision

Key formulas at a glance

Memorise these before attempting numericals — most exam questions hinge on one of them.

Matrix addition

(A+B)_{ij} = a_{ij} + b_{ij}

Multiplication

(AB)_{ij} = \sum_k a_{ik}b_{kj}

Product transpose

(AB)^T = B^T A^T

Inverse

A^{-1} = \frac{\text{adj}(A)}{|A|}

Symmetric decomposition

A = \frac{A+A^T}{2} + \frac{A-A^T}{2}

Product inverse

(AB)^{-1} = B^{-1}A^{-1}

Exam tips

How this chapter is asked

Where this topic appears in CBSE, JEE Main and NEET papers.

AB defined only if columns of A = rows of B; orders follow (m×n)(n×p) = (m×p).
Matrix multiplication is not commutative.
A⁻¹ exists iff |A| ≠ 0; A·A⁻¹ = I.
Aᵀ = A symmetric, Aᵀ = −A skew-symmetric (zero diagonal).
Invert the order in (AB)ᵀ = BᵀAᵀ and (AB)⁻¹ = B⁻¹A⁻¹.
|AB| = |A||B| for square matrices.

FAQ

Common questions

When can two matrices be multiplied?

Only when the number of columns of the first matrix equals the number of rows of the second; the result has the outer dimensions.

When does a matrix have an inverse?

A square matrix has an inverse exactly when its determinant is non-zero, in which case A⁻¹ = adj(A)/|A|.

What is a symmetric matrix?

A square matrix equal to its own transpose (Aᵀ = A). A skew-symmetric matrix satisfies Aᵀ = −A and has a zero main diagonal.

Is AB always equal to BA?

No. Matrix multiplication is not commutative — the orders of the products may even differ, and AB could be defined while BA is not.

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