Inverse Matrix Calculator is a amazing online tool which makes your work easy. It calculates the inverse of matrix of any order if its elements are entered in the given block.

An inverse matrix is a matrix which when multiplied by the original matrix gives the identity matrix. In short it is the reciprocal of the given matrix. It is given as A-1. The inverse of the matrix A is A-1 iff
A $\times$ A-1 = A-1 $\times$ A = I

Inverse of matrix formula is given by
Inverse of Matrix FormulaWhere |A| is the determinant of the matrix and

adj A is the adjoint of the matrix.
To get the inverse of the matrix you are supposed to follow the following steps:

Step 1: Read the given matrix and find the determinant of given matrix and observe whether |A| $\neq$ 0.

Step 2: Find the adjoint of given matrix

Step 3: The inverse of the matrix is given by formula

A-1 = $\frac{1}{|A|}$ adj A
Substituting the values we get the answer.

Below are given some examples based on inverse matrix which may be helpful for you.

Solved Examples

Question 1: Find the inverse of the matrix: A = $\begin{bmatrix} 2& 3\\ 4&1 \end{bmatrix}$
Solution:
 
The given matrix is A = $\begin{bmatrix} 2&3 \\ 4&1 \end{bmatrix}$
Step 1: The determinant of A is given by |A| = 2 - 12 = -10

Step 2: Computation of adj A :
Co-factor of 2 = A11 = 1
Co-factor of 3 = A12 = -4
Co-factor of 4 = A21 = -3
Co-factor of 1 = A22 = 2
The adjoint of A is given by
$\begin{bmatrix}
 1& -4 \\
 -3& 2
\end{bmatrix}$

Step 3: The inverse of matrix is given by
A-1 = $\frac{1}{|A|}$ adj A
      = $\frac{-1}{10}$ $\begin{bmatrix}
 1& -4 \\
 -3& 2
\end{bmatrix}$
     = $\begin{bmatrix}
 \frac{-1}{10}& \frac{4}{10} \\
 \frac{3}{10}& \frac{-2}{10}
\end{bmatrix}$



 

Question 2: Find the inverse of matrix A = $\begin{bmatrix} -1 & 0 & 1 \\ 1 & 2 & 1 \\ 2 & 1 & 2 \end{bmatrix}$
Solution:
 
Given matrix is A = $\begin{bmatrix} -1 & 0 & 1 \\ 1 & 2 & 1 \\ 2 & 1 & 2 \end{bmatrix}$
The determinant is given by |A| = -1(4-1) - 0 + 1 (1-4)
                                             = -3 -3 = -6
To find Adjoint of A Minor of matrix
A11 = (-1)1+1 (4 - 1) = 3
A12 = (-1)1+2 (2 - 2) = 0
A13 = (-1)1+3 (1 - 4) = -3
A21 = (-1)2+1 (0 - 1) = 1
A22 = (-1)2+2 (-2 - 2) = -4
A23 = (-1)2+3 (-1 - 0) = 1
A31 = (-1)3+1 (0 - 2) = -2
A32 = (-1)3+2 (-1 - 1) = 2
A33 = (-1)3+3 (-2 - 0) = -2
The minor is A = $\begin{bmatrix} 3 & 0 & -3 \\ 1 & -4 & 1 \\ -2 & 2 & -2 \end{bmatrix}$
The co-factor is $\begin{bmatrix} 3 & 0 & -3 \\ -1 & -4 & -1 \\ 2 & -2 & 2 \end{bmatrix}$
The adjoint is $\begin{bmatrix} 3 & -1 & 2 \\ 0 & -4 & -2 \\ -3 & -1 & 2 \end{bmatrix}$

Step 3:
The inverse of matrix is A-1 = $\frac{1}{|A|}$ adj A
                                        = $\frac{1}{-6}$ $\begin{bmatrix} 3 & -1 & 2 \\ 0 & -4 & -2 \\ -3 & -1 & 2 \end{bmatrix}$.
                                        = $\frac{1}{6}$ $\begin{bmatrix} -3 & -1 & 2 \\ 0 & 4 & 2 \\ 3 & 1 & -2 \end{bmatrix}$.