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

What must I remember about Transpose and inverse of a matrix?

• Transpose of a Matrix: The matrix A^T obtained by interchanging rows and columns of A; if A is m × n, then A^T is n × m.
• Properties of Transpose: (A^T)^T = A; (A ± B)^T = A^T ± B^T; (kA)^T = k * A^T; Reversal law for product: (AB)^T = B^T * A^T.
• Determinant of 2×2 Matrix: For A = [a b; c d], det(A) = |A| = ad - bc; A is Invertible (Non-singular) if and only if det(A) ≠ 0.
• Inverse of 2×2 Matrix: If det(A) ≠ 0, then A^-1 = (1/det(A)) * [d -b; -c a]; satisfies A * A^-1 = A^-1 * A = I.
• Properties of Inverses: (A^-1)^-1 = A; (kA)^-1 = (1/k) * A^-1; Reversal law for inverse: (AB)^-1 = B^-1 * A^-1; det(A^-1) = 1 / det(A).

Test yourself (1 minute)

1. Singular vs Non-singular

   If det(A) = 0, matrix A is Singular and has NO inverse; if det(A) ≠ 0, A is Non-singular and unique inverse exists.

2. Order of product inverse

   Crucial pitfall: (AB)^-1 is B^-1 * A^-1, NEVER A^-1 * B^-1 due to non-commutativity.

3. Orthogonal Matrix condition

   A square matrix is Orthogonal if A^T * A = I, which implies A^-1 = A^T and det(A) = ±1.

Practise Transpose and inverse of a matrix questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Pure Mathematics, Unit 24

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Year 2 · Term 1