# Inverse matrix of a 2x2 matrix

Show the inverse matrix of a 2x2 matrix
is
, where $|A| = 0$.

### Proof

To find the inverse marix of a 2x2 matrix $A$, we use the property that the inverse matrix is the product of the reciprocal of the determinant and the adjugate matrix, i.e.,
, where $\tilde{A}$ is the adjugate matrix of $A$.
The determinant of $A$ is
(See determinant of a 2x2 matrix.)
The adjugate matrix $\tilde{A}$ is defined as
where $M_{ij}$ is a submatrix obtained by removing i-th row and j-th column from $A$, that is,
and these determinants are
Therefore, each element of $\tilde{A}$ is
and in the matrix form,

By substituing $|A|$ and $\tilde{A}$ into $(1)$, we obtain
Calculator
Enter a 2x2 matrix and press "Execute" button. Then its inverse matrix is displayed.
Matrix
 1 2 1 2
Inverse matrix
 1 2 1 2