For [[a, b], [c, d]], the inverse is (1 / (ad − bc)) times [[d, −b], [−c, a]]. If ad − bc is 0, there is no inverse. Matrices as a set are in matrices.
Swap the diagonal, flip the signs
ad − bc is the determinant, in matrix determinants. Multiplication is a different leftover, in matrix multiplication.
The bottom line
Swap a and d, negate b and c, divide by the determinant. Zero determinant: no inverse.
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