9th Class Math – Chapter 1: Matrices & Determinants

Chapter 1: Matrices & Determinants

Introduction to Matrices

A matrix is a rectangular array of numbers arranged in rows and columns. Matrices are used to solve systems of linear equations and represent linear transformations.

Definition

A matrix of order m × n is written as:

A = [aᵢⱼ] where i = 1, 2, ..., m and j = 1, 2, ..., n

Types of Matrices

**Row Matrix**: A matrix with only one row

Example: [1 2 3]

**Column Matrix**: A matrix with only one column

Example: [1]

[2]

[3]

**Square Matrix**: A matrix with equal number of rows and columns

Example: [1 2]

[3 4]

**Null Matrix**: A matrix with all elements zero

Example: [0 0]

[0 0]

**Identity Matrix**: A square matrix with 1s on the diagonal and 0s elsewhere

Example: [1 0]

[0 1]

Operations on Matrices

#### Addition

Two matrices can be added only if they have the same order. Add corresponding elements.

Example:

[1 2] [5 6] [6 8]

[3 4] + [7 8] = [10 12]

#### Multiplication

Matrix multiplication is possible when the number of columns of the first matrix equals the number of rows of the second matrix.

Determinants

The determinant of a 2×2 matrix [a b; c d] is calculated as:

det = ad - bc

For a 3×3 matrix, we use the expansion method or Sarrus' rule.

Solved Examples

**Example 1**: Find the determinant of [2 3; 1 4]

Solution: det = (2)(4) - (3)(1) = 8 - 3 = 5

**Example 2**: Add the matrices [1 2; 3 4] and [5 6; 7 8]

Solution: [1+5 2+6; 3+7 4+8] = [6 8; 10 12]

Practice Problems

Find the determinant of [3 5; 2 1]

Multiply [1 2; 3 4] by [5; 6]

Find the inverse of [2 1; 3 4]