The formula
A⁻¹ = adj(A) ÷ det(A)
What it means
The inverse undoes what the matrix does. If A transforms a vector, A⁻¹ transforms it back. That only works when the transformation is reversible, which is exactly when the determinant is not zero.
How to do it by hand
- Check the matrix is square
- Work out the determinant: if it is zero, there is no inverse
- Build the matrix of cofactors and transpose it to get the adjugate
- Divide every element by the determinant
What is worth knowing
A zero determinant means the transformation collapses space — it squashes a plane onto a line, say — and you cannot undo that, because the information about which point you started from is gone. Such matrices are called singular. The parallel with numbers is exact: zero has no reciprocal for the same reason. In practice, computing an inverse explicitly is usually the wrong move: to solve Ax = b, elimination is faster and far more numerically stable than working out A⁻¹ and multiplying. Numerical libraries warn about this specifically.