# Matrix Calculator

Matrix Calculator calculates the resultant matrix when certain arithmetic operations are applied to the two given matrices. In mathematics, Matrix is a grid function or a rectangular array in which the numbers are arranged in ordered rows and columns.

## What is Matrix Calculator?

Matrix Calculator is an online tool that helps to perform different matrix operations on 2 × 2 matrices i.e. matrix addition, matrix subtraction, and matrix multiplication. A matrix that has the same number of rows and columns is known as a square matrix. To use this **Matrix calculator**, enter the numbers in the input box.

### Matrix Calculator

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

## How to Use the Matrix Calculator?

Please follow the steps below to find the final matrix using the online matrix calculator:

**Step 1:**Go to Cuemath’s online matrix calculator.**Step 2:**Enter the value of the 2 × 2 matrices in the input boxes and select the operation to be performed from the drop-down list.**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 Calculator Work?

The dimensions of a matrix are usually represented as m x n. Here, m denotes the number of rows while n represents the number of columns in that matrix. Thus, a 2 x 2 matrix will have 2 rows and 2 columns. Subtraction, addition, and multiplication can be performed on matrices. The methods to compute the result for these arithmetic operations are given as follows:

1. **Addition of matrices** - If two matrices have the same number of rows and columns, then addition can be performed. To add two matrices, the elements of each row and column of one matrix are added to the respective elements in the other matrix.

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

2. **Subtraction of matrices** - Similar to addition, we can subtract two matrices only if they have an equal number of rows and columns. We subtract the elements of each row and column of one matrix from the respective elements of the previous matrix.

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

3. **Multiplication of matrices** - To multiply two matrices the number of columns in the first matrix should be equal to the number of rows in the second matrix. Matrix multiplication can be performed as follows:

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

## Solved Examples on Matrix Calculator

**Example 1:**

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

**Solution:**

Matrix 1 + Matrix 2 = \(\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} + \begin{bmatrix} 2 & 1\\ 4 & 2 \end{bmatrix} = \begin{bmatrix} 3 & 3 \\ 7 & 6 \end{bmatrix}\)

**Example 2:**

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

**Solution:**

Matrix 1 - Matrix 2 = \(\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} + \begin{bmatrix} 2 & 1\\ 4 & 2 \end{bmatrix} = \begin{bmatrix} -1 & 1 \\ -1 & 2 \end{bmatrix}\)

Similarly, you can try the matrix calculator to add, subtract and multiply the following matrices:

- Matrices = \(\begin{bmatrix} 2 & 1 \\ 4 & 2 \end{bmatrix}\) & \(\begin{bmatrix} 3 & 3 \\ 7 & 6 \end{bmatrix}\)
- Matrices = \(\begin{bmatrix} 5 & 8 \\ 10 & 16 \end{bmatrix}\) & \(\begin{bmatrix} 8 & 2 \\ 1 & 3 \end{bmatrix}\)

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