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 \(\det.(adj.\, A^{-1})\) 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 X = A⁻¹B if and only if A is non-singular (det(A) ≠ 0); understanding consistency conditions and rank relationships determines existence and uniqueness of solutions across all cases.
<p><strong>Step 1:</strong> For system AX = B, check three conditions:</p><p>• <strong>Unique solution:</strong> det(A) ≠ 0 (A is non-singular) → X = A⁻¹B exists uniquely</p><p>• <strong>Infinite solutions:</strong> rank(A) = rank([A|B]) < order of A (consistent, dependent)</p><p>• <strong>No solution:</strong> rank(A) < rank([A|B]) (inconsistent)</p><p><strong>Step 2:</strong> Verify each option:</p><p>• If det(A) ≠ 0: unique solution ✓</p><p>• If rank(A) = rank([A|B]) = n: unique solution ✓</p><p>• If rank(A) = rank([A|B]) < n: infinitely many solutions ✓</p><p>• If rank(A) < rank([A|B]): no solution (inconsistent) ✗</p><p>∴ Answer: A,B,C (assuming these represent the first three conditions listed above)</p>
Correct Answer: A,B,C