1.4.3—Inverse matrix products

If A is invertible, AA⁻¹=A⁻¹A=I. To solve AX=B, multiply on the left by A⁻¹: X=A⁻¹B. The side matters because matrix multiplication is not generally commutative.

For a product, (AB)⁻¹=B⁻¹A⁻¹: the operations are undone in reverse order. Check dimensions and multiply back to verify the result.

If a transformation first applies B and then A, its matrix is AB; reversing the process uses B⁻¹A⁻¹, not A⁻¹B⁻¹.

Do not divide by a matrix as if it were a scalar, and do not move A⁻¹ across B without changing the order.