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Inverse Matrix Calculator
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Inverse matrix calculator is an online tool to calculate the inverse matrix value of given 2x2, 3x3, 4x4 matrix input values.

All square matrices will not have inverses. A square matrices which has an inverses is known as invertible or non singular, and a square matrix which does not have an inverse is known as non-invertible or singular.

You can see a default 2x2 matrix given below. Click on "Submit", the calculator first calculates determinant and adjoint of the matrix. Then, finds the transpose of the matrix and by interchanging rows n column, inverse of the matrix is calculated by using the formula $\frac{adj A}{|A|}$.

## Step by Step Calculation

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Step 1 :

Find the determinant of a matrix that is |A|.

Step 2 :

Find the adjoint of A, transpose after getting the adjoint.

Step 3 :

Inverse A, that is A-1 = $\frac{adj A}{|A|}$ ; |A| $\neq$ 0

## Example Problems

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1. ### If A = $\begin{bmatrix}5& 6\\ -7& 8 \end{bmatrix}$. Find A-1.

Step 1 :

Given, A = $\begin{bmatrix}5& 6\\ -7& 8 \end{bmatrix}$

|A| = 40 + 42 = 82

Step 2 :

adj A = $\begin{bmatrix}8& -7\\ 6& 5 \end{bmatrix}$, Transponse = $\begin{bmatrix}8& 7\\ -6& 5 \end{bmatrix}$

Inter changing row and column

adj A = $\begin{bmatrix}8& -6\\ 7& 5 \end{bmatrix}$

Step 3 :

A-1 = $\frac{\begin{bmatrix}8& -6\\ 7& 5 \end{bmatrix}}{82}$

A-1 = $\frac{1}{82}$ $\begin{bmatrix}8& -6\\ 7& 5 \end{bmatrix}$.

Answer  :

A-1 = $\frac{1}{82}$ $\begin{bmatrix}8& -6\\ 7& 5 \end{bmatrix}$

2. ### If A = $\begin{bmatrix}1 & 0 & 3\\ 2 & 1 & 2\\ 0 & 0 & -1\end{bmatrix}$ . Find A-1.

Step 1 :

Given, $\begin{bmatrix}1 & 0 & 3\\ 2 & 1 & 2\\ 0 & 0 & -1\end{bmatrix}$

A-1 = $\frac{adj A}{|A|}$

Step 2 :

adj A = $\begin{bmatrix}-1 & -2 & 0\\ 0 & -1 & 0\\ -3 & -4 & 1\end{bmatrix}$,

transpose = $\begin{bmatrix}-1 & 2 & 0\\ 0 & -1 & 0\\ -3 & 4 & 1\end{bmatrix}$

Inter changing rows and columns

adj A = $\begin{bmatrix}-1 & 0 & -3\\ 2 & -1 & 4\\ 0 & 0 & 1\end{bmatrix}$

Step 3 :

|A| = 1(-1) - (-2) + 3(0) = -1 + 2 = 1.

A-1 = $\frac{\begin{bmatrix}-1 & 0 & -3\\ 2 & -1 & 4\\ 0 & 0 & 1\end{bmatrix}}{1}$

A-1 = $\begin{bmatrix}1 & 0 & 3\\ -2 & 1 & -4\\ 0 & 0 & -1\end{bmatrix}$

Answer  :

A-1 = $\begin{bmatrix}1 & 0 & 3\\ -2 & 1 & -4\\ 0 & 0 & -1\end{bmatrix}$

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