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, use row reduction on ([A mid I])
Apply elementary row operations until the left side becomes the identity matrix
The right side becomes the inverse matrix
Alternatively, use the adjugate formula (A^{-1}=frac{1}{det(A)}operatorname{adj}(A))
Verify the result by multiplying (A cdot A^{-1} = I)