Check that the matrix is square
Compute the determinant
If the determinant is 0, the matrix has no inverse
For a 2×2 matrix (begin{bmatrix} a & b \ c & d end{bmatrix}), use (frac{1}{ad-bc}begin{bmatrix} d & -b \ -c & a end{bmatrix})
For larger matrices, form the augmented matrix ([A mid I])
Use row operations to transform (A) into the identity matrix
Apply the same row operations to the identity matrix
The transformed identity matrix becomes (A^{-1})
Alternatively, compute the adjugate matrix and divide by the determinant
Verify the result by checking that (AA^{-1}=I) and (A^{-1}A=I)