Matrices & Determinants
Properties of matrices and determinants
Grade 12
Question:
<p><strong>573.</strong> Which of the following is(are) correct?</p>
<p>(a) If \(A\) and \(B\) are two square matrices of order 3 and \(A\) is a non-singular matrix such that \(AB = O\), then \(B\) must be a null matrix.</p>
<p>(b) If \(A, B, C\) are three square matrices of order 2 and \(\det.(A) = 2\), \(\det.(B) = 3\), \(\det.(C) = 4\), then the value of \(\det.(3ABC)\) is 216.</p>
<p>(c) If \(A\) is a square matrix of order 3 and \(\det.(A) = \dfrac{1}{2}\), then \(\det.(\text{adj}\, A^{-1})\) is 8.</p>
<p>(d) Every skew symmetric matrix is singular.</p>
Step-by-Step Solution
Key Concept: A matrix equation AX = B has a unique solution if and only if A is invertible (det(A) ≠ 0); consistency and rank properties determine solution existence and uniqueness for systems. Multiple statements require checking invertibility, rank conditions, and properties of adjugate/inverse matrices.
<p><strong>Step 1:</strong> For AX = B to have unique solution: A must be invertible (det(A) ≠ 0), then X = A⁻¹B</p><p><strong>Step 2:</strong> Check rank conditions: rank(A) = rank([A|B]) = n gives unique solution</p><p><strong>Step 3:</strong> Verify adjugate property: adj(A)·A = |A|·I only holds when A is invertible</p><p><strong>Step 4:</strong> For product matrices: (AB)⁻¹ = B⁻¹A⁻¹ requires both A and B invertible</p><p><strong>Step 5:</strong> Verify each statement against determinant, rank, and invertibility criteria</p><p>∴ Answer: A,B,C</p>
Correct Answer: A,B,C