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:

A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}, B = \begin{bmatrix} b_{11}\\ b_{21}\\ b_{31} \end{bmatrix}

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

C = A \times B = \begin{bmatrix} c_{11} \\ c_{21} \\ c_{31} \end{bmatrix} = \begin{bmatrix} a_{11}b_{11} + a_{12}b_{21} + a_{13}b_{31} \\ a_{21}b_{11} + a_{22}b_{21} + a_{23}b_{31} \\ a_{31}b_{11} + a_{32}b_{21} + a_{33}b_{31} \end{bmatrix}

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 A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}, B = \begin{bmatrix} 1\\ 2\\ 3 \end{bmatrix}, then find C?

Solution:

Calculation of the product C:

C = A \times B = \begin{bmatrix} 1\cdot1 + 2\cdot2 + 3\cdot3 \\ 4\cdot1 + 5\cdot2 + 6\cdot3 \\ 7\cdot1 + 8\cdot2 + 9\cdot3 \\ \end{bmatrix}

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

C = \begin{bmatrix} 14 \\ 32 \\ 50 \end{bmatrix}

Example 2: Let X = \begin{bmatrix} -9 & -8 & -7 \\ -6 & -5 & -4 \\ -3 & -2 & -1 \end{bmatrix}, Y = \begin{bmatrix} -3\\ -2\\ -1 \end{bmatrix}, then find Z?

Solution:

Calculation of the product Z:

Z = X \times Y = \begin{bmatrix} (-9)\cdot(-3) & (-8)\cdot(-2) & (-7)\cdot(-1)\\ (-6)\cdot(-3) & (-5)\cdot(-2) & (-4)\cdot(-1)\\ (-3)\cdot(-3) & (-3)\cdot(-2) & (-1)\cdot(-1)\\ \end{bmatrix}

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

Z = \begin{bmatrix} 50 \\ 32 \\ 24 \end{bmatrix}

Example 3: Let X = \begin{bmatrix} 11 & 22 & -33 \\ -44 & 55 & 66 \\ 77 & -88 & 99 \end{bmatrix}, Y = \begin{bmatrix} 9\\ -8\\ -7 \end{bmatrix}, find Z?

Solution:

Calculation of the product Z:

Z = X \times Y = \begin{bmatrix} (11)\cdot(9) & (22)\cdot(-8) & (-33)\cdot(7)\\ (-44)\cdot(9) & (55)\cdot(-8) & (66)\cdot(7)\\ (77)\cdot(9) & (-88)\cdot(-8) & (99)\cdot(7)\\ \end{bmatrix}

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

Z = \begin{bmatrix} -308 \\ -374 \\ 2090 \end{bmatrix}

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