Matrix InverseCalculator

Matrix Inverse

The inverse of a matrix A is another matrix, denoted as A^-1, such that when A is multiplied by A^-1, the result is the identity matrix. The matrix A must be square (same number of rows and columns) and have a non-zero determinant to have an inverse.

The inverse of a matrix A is computed as:

$A^{-1}=\frac{1}{det(A)}\cdot \text{adj}(A)$

where $det(A)$ is the determinant of A and $\text{adj}(A)$ is the adjugate of A.

Example 1

Let's find the inverse of the following 2x2 matrix:

$A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$

The determinant of A is:

$det(A)=(1\times 4)-(2\times 3)=4-6=-2$

The adjugate of A is:

$\text{adj}(A)=\left[\begin{array}{cc}4& -2\\ -3& 1\end{array}\right]$

Therefore, the inverse of A is:

$A^{-1}=\frac{1}{-2}\times \left[\begin{array}{cc}4& -2\\ -3& 1\end{array}\right]=\left[\begin{array}{cc}-2& 1\\ 1.5& -0.5\end{array}\right]$