Transpose and inverse of a matrix — A/L Combined Mathematics

The transpose of an m × n matrix A, denoted by A^T, is the n × m matrix formed by interchanging the rows and columns of A. Key transpose identities include: (A^T)^T = A, (A + B)^T = A^T + B^T, (kA)^T = k A^T, and the reversal law for products: (AB)^T = B^T A^T. The inverse of a square matrix A is a matrix A⁻¹ satisfying AA⁻¹ = A⁻¹A = I. An inverse exists if and only if A is non-singular, meaning det(A) ≠ 0. For a 2 × 2 matrix A = [[a, b], [c, d]], the inverse formula is: A⁻¹ = [1 / (ad - bc)] [[d, -b], [-c, a]]. The inverse also obeys the reversal law: (AB)⁻¹ = B⁻¹ A⁻¹.

Check before you answer (A/L Pure Maths Paper 1)

1. Product Reversal Law Application: Applying (AB)^T = B^T A^T and (AB)⁻¹ = B⁻¹ A⁻¹ in theoretical matrix proofs.
2. Inverse Formula Execution: Swapping main diagonal entries (a and d), negating off-diagonal entries (-b and -c), and dividing by the non-zero determinant ad - bc.
3. Verification of Invertibility: Multiplying the candidate A⁻¹ by A to explicitly confirm that the product yields the 2 × 2 identity matrix I.