How to Multiply 3 × 3 Matrices

Matrix multiplication is a mathematical operation performed on two matrices that results in a single matrix, called the product matrix. It combines two matrices by following specific rules, where each element of the product matrix is derived from the dot product of the corresponding rows of the first matrix and the columns of the second matrix.

Let's say we have two 3 × 3 matrices A and B:

and

The result of multiplying two 3 × 3 matrices is also a 3 × 3 matrix, denoted as C:

Each element c_ij in the resulting matrix C is computed as the dot product of the i^th row of matrix A and the j^th column of matrix B.

The formula for each element is:

c_ij = a_i1 · b_1j + a_i2 · b_2j + a_i3 · b_3j

Where i and j are the row and column indices, respectively.

Note: Jacques Philippe Marie Binet was the first to perform matrix multiplication in 1812.

Solved Examples on 3 × 3 Matrix:

Example 1: Let and , then find C?

Solution:

Calculation of the product C

Thus the product of the matrix A and B is:

.

Example 2: Let and , then find C?

Solution:

Calculation of the product C

Thus the product of the matrix A and B is:

Example 3: Given Matrix A =. Calculate A².

Solution:

To find A², we multiply matrix A by itself:

A² = A × A

Read More,

• Matrix Multiplication
• Matrix Addition
• Operation on Matrices
• Elementary Operations on Matrices