Class 12 Maths Matrices Notes

3.1 Introduction

Matrices are an important tool in mathematics used to arrange data in rows and columns. They are widely applied in science, engineering, economics, and computer graphics. In Class 12, matrices help in solving systems of linear equations and understanding advanced algebraic concepts.

3.2 Matrix

A matrix is a rectangular array of numbers or variables arranged in rows and columns, enclosed within brackets.

Example: $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ A=[1324]

The horizontal lines are called rows
The vertical lines are called columns
A matrix with m rows and n columns is called a matrix of order m × n

3.3 Types of Matrices

🔹 1. Row Matrix

A matrix having only one row.

🔹 2. Column Matrix

A matrix having only one column.

🔹 3. Rectangular Matrix

A matrix in which number of rows ≠ number of columns.

🔹 4. Square Matrix

A matrix in which number of rows = number of columns.

🔹 5. Zero (Null) Matrix

A matrix in which all elements are zero.

🔹 6. Diagonal Matrix

A square matrix in which all non-diagonal elements are zero.

🔹 7. Scalar Matrix

A diagonal matrix in which all diagonal elements are equal.

🔹 8. Identity Matrix

A scalar matrix in which all diagonal elements are 1.
It is denoted by I.

3.4 Operations on Matrices

🔹 1. Addition of Matrices

Two matrices can be added only if they have the same order. $A + B = [a_{ij} + b_{ij}]$ A+B=[aij+bij]

🔹 2. Subtraction of Matrices

Subtraction is possible only for matrices of the same order.

🔹 3. Multiplication of a Matrix by a Scalar

Each element of the matrix is multiplied by a number.

🔹 4. Multiplication of Two Matrices

Matrix multiplication is possible only when
Number of columns of first matrix = Number of rows of second matrix
Matrix multiplication is not commutative
(A × B ≠ B × A)

3.5 Transpose of a Matrix

The transpose of a matrix is obtained by interchanging rows and columns.

If A is a matrix, then its transpose is denoted by Aᵀ.

Properties of Transpose:

(Aᵀ)ᵀ = A
(A + B)ᵀ = Aᵀ + Bᵀ
(AB)ᵀ = BᵀAᵀ

3.6 Symmetric and Skew Symmetric Matrices

🔹 Symmetric Matrix

A square matrix A is symmetric if: $Aᵀ = A$ AT=A

🔹 Skew Symmetric Matrix

A square matrix A is skew symmetric if: $Aᵀ = -A$ AT=−A

Diagonal elements of a skew symmetric matrix are always zero

3.7 Invertible Matrices

A square matrix A is said to be invertible if there exists a matrix A⁻¹ such that: $AA^{-1} = A^{-1}A = I$ AA−1=A−1A=I

Conditions for Invertibility:

Matrix must be square
Determinant of A ≠ 0

Important Formula:

$A^{-1} = \frac{1}{|A|} \, \text{Adj}(A)$ A−1=∣A∣1Adj(A)

✅ Key Points to Remember

Matrix multiplication is not commutative
Only square matrices can be invertible
Identity matrix acts like 1 in matrix multiplication
Symmetric and skew symmetric matrices are always square matrices