Matrices & Determinants
Properties of Determinants and Matrices
Grade 12
Question:
<p>Which of the following statements is/are correct?<br>(a) If <i>AB = O</i>, where det(<i>A</i>) ≠ 0, then <i>B</i> = null matrix.<br>(b) If det(<i>A</i>) = 2, det(<i>B</i>) = 3, det(<i>C</i>) = 4, and <i>A</i>, <i>B</i>, <i>C</i> are square matrices of order 2, then det(3<i>ABC</i>) = 216.<br>(c) If det(<i>A</i>) = \(\frac{1}{2}\) (order of matrix <i>A</i> is 3), then det(adj. \(A^{-1}\)) = 4.<br>(d) A skew symmetric matrix of even order need not be singular.</p>
<p>(a)</p>
<p>(b)</p>
<p>(c)</p>
<p>(d)</p>
Step-by-Step Solution
Key Concept: Use the properties: (1) If det(A) ≠ 0, then A is invertible, so AB = O implies B = O; (2) det(kA) = k^n·det(A) for n×n matrix; (3) For adj(A^-1), use det(adj(M)) = [det(M)]^(n-1) and properties of inverses.
<p><strong>Statement (a):</strong> If AB = O and det(A) ≠ 0, then A is invertible. Multiply both sides by A⁻¹: A⁻¹(AB) = A⁻¹O ⟹ B = O. <strong>✓ CORRECT</strong></p><p><strong>Statement (b):</strong> det(3ABC) = 3²·det(A)·det(B)·det(C) = 9·2·3·4 = 216. (For 2×2 matrix, scalar k gives k² factor) <strong>✓ CORRECT</strong></p><p><strong>Statement (c):</strong> For det(A) = 1/2 (order 3): det(A⁻¹) = 1/det(A) = 2. For det(adj(M)), use formula: det(adj(M)) = [det(M)]^(n-1). So det(adj(A⁻¹)) = [det(A⁻¹)]² = [2]² = 4. <strong>✓ CORRECT</strong></p><p><strong>Statement (d):</strong> For skew-symmetric matrix A (of any order): det(A) = det(Aᵀ) = det(-A) = (-1)ⁿdet(A). For <em>even</em> order (n even), this gives det(A) = det(A), which is always true—A need not be singular. Example: 2×2 skew-symmetric [[0, 1], [-1, 0]] has det = 1 ≠ 0. <strong>✗ INCORRECT</strong></p><p><strong>Answer: (a), (b)</strong></p>
Correct Answer: a,b