Inverse matrices

For a 2 × 2 matrix A = [[a,b],[c,d]] with determinant Δ ≠ 0, A⁻¹ = [[d,−b],[−c,a]]/Δ. State the nonzero determinant before applying the formula.

Check an inverse by multiplying to obtain the identity. For a product, (AB)⁻¹ = B⁻¹A⁻¹; reversing the order undoes the transformations in reverse sequence. An inverse of a matrix is different from reciprocating each entry.

Worked example

Find the inverse of [[2,1],[1,1]].

1. The determinant is 2 × 1 − 1 × 1 = 1.
2. Swap the diagonal entries and negate the other entries.
3. Multiplication with the original gives [[1,0],[0,1]].

Answer: [[1,−1],[−1,2]]

Revise first: Determinants and singular matrices.