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⇱ How to Multiply a 3 × 3 Matrix with a 3 × 1 Matrix - GeeksforGeeks

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How to Multiply a 3 × 3 Matrix with a 3 × 1 Matrix

Last Updated : 23 Jul, 2025

Matrix multiplication is a way to combine two matrices (rectangular arrays of numbers) to create a new matrix.

  • You can only multiply two matrices if the number of columns in the first matrix matches the number of rows in the second matrix.
  • The resulting matrix will have dimensions equal to the number of rows of the first matrix and the number of columns of the second matrix.

For example, if you have a matrix A that is 2 × 3 (2 rows and 3 columns) and a matrix B that is 3 × 2 (3 rows and 2 columns), you can multiply them. Result here will be a 2 × 2 matrix.

To find each element in the resulting matrix, follow these steps:

  • Take a row from the first matrix.
  • Take a column from the second matrix.
  • Multiply corresponding elements from the row and column together and then add those products.

Matrix multiplication of the 3 × 3 Matrix by 3 × 1 Matrix

Let A be a 3 × 3 Matrix and B be a 3 × 1 matrix:

The resulting matrix C, which will be a 3 × 1 matrix, is given by:

Solved Examples on 3 × 3 Matrix by 3 × 1 Matrix

Let's consider some examples of matrix multiplication 3 × 3 Matrix by 3 × 1 Matrix:

Example 1: Let , then find C?

Solution:

Calculation of the product C:

Thus, the product of the matrix A and B is:

Example 2: Let , then find Z?

Solution:

Calculation of the product Z:

Thus, the product of the matrix X and Y is:

Example 3: Let , find Z?

Solution:

Calculation of the product Z:

Thus, the product of the matrix X and Y is:

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