# Inverse Matrix Calculator

Inverse Matrix Calculator calculates the value of the inverse matrix for a given matrix. Suppose we multiply a matrix with a given matrix and the product is the multiplicative identity. Then such a matrix is known as the inverse of the given matrix.

## What is Inverse Matrix Calculator?

Inverse Matrix Calculator is an online tool that helps to compute the inverse matrix for a given 2 × 2 matrix. If the determinant of a matrix is 0 then the inverse of such a matrix cannot exist. To use this * inverse matrix calculator*, enter the values in the input boxes.

### Inverse Matrix Calculator

*Use up to 3 digits only.

## How to Use the Inverse Matrix Calculator?

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

**Step 1:**Go to Cuemath’s online inverse matrix calculator.**Step 1:**Enter the value of the matrix in the given input boxes of the inverse matrix calculator.**Step 2:**Click on the**"Calculate"**button to find the resultant inverse matrix.**Step 3:**Click on the**"Reset"**button to clear the fields and enter new values.

## How Does Inverse Matrix Calculator Work?

Suppose we have a matrix given by A. We denote the inverse by A^{-1}. When these two matrices are multiplied we get the multiplicative identity. Thus, A.A^{-1} = I. For the inverse of a matrix to exist, its determinant should not be 0. Further, the matrix needs to be a square matrix. The steps to calculate the inverse of a 2 × 2 matrix are as follows:

- Step 1: Find the determinant of the given matrix.
- Step 2: Find the cofactor matrix.
- Step 3: Find the transpose of the cofactor matrix. This is also known as the adjoint of the matrix.
- Step 4: Divide the adjoint of the matrix obtained in step 3 by the determinant from step 1. This will give the inverse of the matrix.

In addition to the steps mentioned above, the following formula can also be used to find the inverse of a 2 × 2 matrix.

A = \(\begin{bmatrix} a & b\\ c & d \end{bmatrix}\)

A^{-1} = \(\frac{1}{\begin{vmatrix} A \end{vmatrix}}adjA\)

A^{-1} = \(\frac{1}{ad - bc}\begin{bmatrix} d & -b\\ -c & a \end{bmatrix}\)

## Solved Examples on Inverse Matrix

**Example 1:** Find the inverse matrix of A = \(\begin{bmatrix} 1 & 2\\ 3 & 4 \end{bmatrix}\) and verify it using the inverse matrix calculator.

**Solution:**

Given: A = \(\begin{bmatrix} 1 & 2\\ 3 & 4 \end{bmatrix}\)

A^{-1} = \(\frac{1}{ad - bc}\begin{bmatrix} d & -b\\ -c & a \end{bmatrix}\)

A^{-1} = \(\frac{1}{4 - 6}\begin{bmatrix} 4 & -2\\ -3 & 1 \end{bmatrix}\)

A^{-1 }= \(\begin{bmatrix} -2 & 1\\ 1.5 & -0.5 \end{bmatrix}\)

**Example 2:** Find the inverse matrix of A = \(\begin{bmatrix} -10 & 4\\ 3 & -1 \end{bmatrix}\) and verify it using the inverse matrix calculator.

**Solution:**

Given: A = \(\begin{bmatrix} -10 & 4\\ 3 & -1 \end{bmatrix}\)

A^{-1} = \(\frac{1}{ad - bc}\begin{bmatrix} d & -b\\ -c & a \end{bmatrix}\)

A^{-1} = \(\frac{1}{10 - 12}\begin{bmatrix} -1 & -4\\ -3 & -10 \end{bmatrix}\)

A^{-1 }= \(\begin{bmatrix} 0.5 & 2\\ 1.5 & 5 \end{bmatrix}\)

Similarly, you can try the inverse matrix calculator and find the inverse for the given matrices:

- A = \(\begin{bmatrix} 5 & 10\\ 15 & 20 \end{bmatrix}\)
- B = \(\begin{bmatrix} 18 & 21\\ 7 & 3 \end{bmatrix}\)

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