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   Enter a 2x2 matrix and press "Execute" button. Then its inverse matrix is displayed.