Check that the matrix is square
Verify that the determinant is nonzero
Use the formula for a 2×2 matrix: if (A=begin{bmatrix}a&b\c&dend{bmatrix}), then (A^{-1}=frac{1}{ad-bc}begin{bmatrix}d&-b\-c&aend{bmatrix})
Use row reduction on the augmented matrix ([A mid I])
Perform elementary row operations until the left side becomes the identity matrix
Read the inverse from the right side of the augmented matrix
Use the adjugate formula: (A^{-1}=frac{1}{det(A)}operatorname{adj}(A))
Use Gaussian elimination or Gauss-Jordan elimination for larger matrices
Use a calculator or software for complex matrices
Confirm the result by multiplying (A) and (A^{-1}) to get the identity matrix