Matrices & Determinants
Properties of Matrices
Grade 12
Question:
<p>For \(3 \times 3\) matrices \(M\) and \(N\), which of the following statement(s) is(are) NOT correct?</p><p>(1) \(N^T MN\) is symmetric or skew-symmetric, according as \(M\) is symmetric or skew-symmetric</p><p>(2) \(MN - NM\) is skew-symmetric for all symmetric matrices \(M\) and \(N\)</p><p>(3) \(MN\) is symmetric for all symmetric matrices \(M\) and \(N\)</p><p>(4) \((\text{adj}\, M)(\text{adj}\, N) = \text{adj}(MN)\) for all invertible matrices \(M\) and \(N\)</p>
<p>\(N^T MN\) is symmetric or skew-symmetric, according as \(M\) is symmetric or skew-symmetric</p>
<p>\(MN - NM\) is skew-symmetric for all symmetric matrices \(M\) and \(N\)</p>
<p>\(MN\) is symmetric for all symmetric matrices \(M\) and \(N\)</p>
<p>\((\text{adj}\, M)(\text{adj}\, N) = \text{adj}(MN)\) for all invertible matrices \(M\) and \(N\)</p>
Step-by-Step Solution
Key Concept: Test each statement by checking definitions: a matrix A is symmetric if A^T = A, skew-symmetric if A^T = -A, and verify whether properties hold universally or only under specific conditions. The key is recognizing that some statements claim 'for all' matrices when counterexamples exist.
<p><strong>Statement (1):</strong> If M is symmetric, then (N^T MN)^T = N^T M^T N = N^T MN ✓ (symmetric). If M is skew-symmetric, then (N^T MN)^T = N^T M^T N = -N^T MN ✓ (skew-symmetric). <strong>CORRECT</strong></p><p><strong>Statement (2):</strong> For symmetric M and N: (MN - NM)^T = N^T M^T - M^T N^T = NM - MN = -(MN - NM) ✓ (skew-symmetric for all symmetric matrices M, N). <strong>CORRECT</strong></p><p><strong>Statement (3):</strong> For symmetric M and N: (MN)^T = N^T M^T = NM. But MN ≠ NM in general (matrices don't commute). <strong>COUNTEREXAMPLE:</strong> Let M = [[1,0],[0,0]] and N = [[0,1],[0,0]]. Then MN = [[0,1],[0,0]] but NM = [[0,0],[0,0]]. MN is NOT symmetric. <strong>INCORRECT</strong></p><p><strong>Statement (4):</strong> adj(MN) = adj(N)adj(M), NOT adj(M)adj(N). The correct order is reversed. <strong>INCORRECT</strong></p><p>∴ Answer: CD (Statements 3 and 4 are NOT correct)</p>
Correct Answer: CD