MATRIX METHOD GUIDE
Matrix inverse and determinant: when can you use them?
The determinant indicates whether a square matrix is invertible. An inverse exists only when the determinant is not zero.FORMULA
For a 2×2 matrix [[a, b], [c, d]], det(A) = ad − bcWORKED EXAMPLE
If det(A) = 0, the rows or columns are dependent and A⁻¹ does not exist. Use RREF to investigate the system instead.STEP BY STEP
Check that the matrix is square
Determinants and ordinary inverses apply only to square matrices with the same number of rows and columns.
Calculate the determinant
A non-zero determinant means the matrix is invertible and a unique inverse exists.
Find the inverse when valid
Gauss-Jordan elimination can reduce [A | I] into [I | A⁻¹].
Use the inverse for a system
For A · x = b, multiply A⁻¹ by b only when A is square and invertible.