# Matrix Multiplication Calculator

Matrix Multiplication Calculator calculates the product of the two given matrices. A matrix is a rectangular array or a grid in which numbers are arranged in rows and columns.

## What is a Matrix Multiplication Calculator?

Matrix Multiplication Calculator is an online tool that helps to perform multiplication on matrices. Matrices are widely used to represent data while working with linear equations, geometry, and statistics. To use this **matrix multiplication calculator**, enter the values in the input boxes

### Matrix Multiplication Calculator

**NOTE:** Enter upto 3 digits only.

## How to Use the Matrix Multiplication Calculator?

Please follow the steps below to find the product of matrices using the online matrix multiplication calculator:

**Step 1:**Go to Cuemath’s online matrix multiplication calculator.**Step 2:**Choose the dimensions of the matrices ( 2 x 2 or 3 x 3) from the drop-down list and enter the values in the matrix multiplication calculator.**Step 3:**Click on the**"Calculate"**button to find the resultant matrix.**Step 4:**Click on the**"Reset"**button to clear the fields and enter new values.

## How does Matrix Multiplication Calculator Works?

A matrix that has m rows and n columns is represented as \(A_{m \times n}\). This is called a rectangular matrix. Further, if a matrix has the same number of rows and columns it is called a square matrix. For example, a 2 x 2 matrix will be a square matrix as it has 2 rows and 2 columns. To multiply two matrices, the number of columns of the first matrix should be the same as the number of rows of the second matrix. If this condition is not satisfied, matrix multiplication cannot be performed. Thus, if we have two matrices with the dimensions 5 x 3 and 3 x 2 respectively, then they can be multiplied. However, a 6 x 1 matrix cannot be multiplied by a 2 x 4 matrix. Given below is the procedure to perform matrix multiplication on 2 x 2 and 3 x 3 matrices.

1. **2 × 2 Matrices**

A x B = \(\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} \times \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix} =\begin{bmatrix} a_{11}b_{11} + a_{12}b_{21} & a_{11}b_{12} + a_{12}b_{22}\\ a_{21}b_{11} + a_{22}b_{21} & a_{21}b_{12} + a_{22}b_{22} \end{bmatrix}\)

2. **3 × 3 Matrices**

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

= \(\begin{bmatrix} a_{11}b_{11} + a_{12}b_{21} + a_{13}b_{31} & a_{11}b_{12} + a_{12}b_{22} + a_{13}b_{32} & a_{11}b_{13} + a_{12}b_{23} + a_{13}b_{33} \\ a_{21}b_{11} + a_{22}b_{21} + a_{23}b_{31} & a_{21}b_{12} + a_{22}b_{22} + a_{23}b_{32} & a_{21}b_{13} + a_{22}b_{23} + a_{23}b_{33}\\ a_{31}b_{11} + a_{32}b_{21} + a_{33}b_{31} & a_{31}b_{12} + a_{32}b_{22} + a_{33}b_{32} & a_{31}b_{13} + a_{32}b_{23} + a_{33}b_{33} \end{bmatrix}\)

## Solved Examples on Matrix Multiplication Calculator

**Example 1:**

Multiply the matrices \(\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\) & \(\begin{bmatrix} 2 & 1 \\ 4 & 2 \end{bmatrix}\)** **and verify it using the matrix multiplication calculator.

**Solution:**

Multiplication = \(\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} \times \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix} =\begin{bmatrix} a_{11}b_{11} + a_{12}b_{21} & a_{11}b_{12} + a_{12}b_{22}\\ a_{21}b_{11} + a_{22}b_{21} & a_{21}b_{12} + a_{22}b_{22} \end{bmatrix}\)

Multiplication = \(\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \times \begin{bmatrix} 2 & 1\\ 4 & 2 \end{bmatrix} = \begin{bmatrix} 10 & 5 \\ 22 & 11 \end{bmatrix}\)

**Example 2 :**

Multiply the matrices \(\begin{bmatrix} 3 & 2 & 5\\ 5 & 2 & 4 \\ 2 & 5 & 6 \end{bmatrix}\) & \(\begin{bmatrix} 2 & 2 & 7\\ 1 & 4 & 6 \\ 3 & 8 & 7 \end{bmatrix}\)** **and verify it using the matrix multiplication calculator.

**Solution:**

Multiplication = \(\begin{bmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} \times \begin{bmatrix} b_{11} & b_{12} & b_{13}\\ b_{21} & b_{22} & b_{23} \\ b_{31} & b_{32} & b_{33} \end{bmatrix}\)

= \(\begin{bmatrix} a_{11}b_{11} + a_{12}b_{21} + a_{13}b_{31} & a_{11}b_{12} + a_{12}b_{22} + a_{13}b_{32} & a_{11}b_{13} + a_{12}b_{23} + a_{13}b_{33} \\ a_{21}b_{11} + a_{22}b_{21} + a_{23}b_{31} & a_{21}b_{12} + a_{22}b_{22} + a_{23}b_{32} & a_{21}b_{13} + a_{22}b_{23} + a_{23}b_{33}\\ a_{31}b_{11} + a_{32}b_{21} + a_{33}b_{31} & a_{31}b_{12} + a_{32}b_{22} + a_{33}b_{32} & a_{31}b_{13} + a_{32}b_{23} + a_{33}b_{33} \end{bmatrix}\)

Multiplication = \(\begin{bmatrix} 3 & 2 & 5\\ 5 & 2 & 4 \\ 2 & 5 & 6 \end{bmatrix} \times \begin{bmatrix} 2 & 2 & 7\\ 1 & 4 & 6 \\ 3 & 8 & 7 \end{bmatrix}\)

= \(\begin{bmatrix} 23 & 54 & 68\\ 24 & 50 & 75 \\ 27 & 72 & 86 \end{bmatrix}\)

Similarly, you can try the matrix multiplication calculator and multiply the following matrices.

- Matrices = \(\begin{bmatrix} 5 & 7 & 2\\ 1 & 8 & 10 \\ 11 & 18 & 1 \end{bmatrix}\) , \(\begin{bmatrix} 3 & 7 & 8\\ 1 & 7 & 1 \\ 5 & 19 & 1 \end{bmatrix}\)
- Matrices = \(\begin{bmatrix} 4 & 8 \\ 12 & 16 \end{bmatrix}\), \(\begin{bmatrix} 5 & 10 \\ 15 & 20 \end{bmatrix}\)

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