<p>Let \(M\) and \(N\) be two \(3 \times 3\) nonsingular skew-symmetric matrices such that \(MN = NM\). If \(P^T\) denotes the transpose of \(P\), then \(M^2 N^2 (M^T N)^{-1} (MN^{-1})^T\) is equal to</p>
Step-by-Step Solution
Key Concept: For skew-symmetric matrices A: A^T = -A, so A^2 is symmetric negative semi-definite. Use commutativity MN = NM and properties of transposes with skew-symmetric matrices to simplify the expression systematically.
<p><strong>Step 1: Apply skew-symmetric properties</strong></p><p>Since M and N are skew-symmetric: M^T = -M and N^T = -N</p><p><strong>Step 2: Simplify (M^T N)^-1</strong></p><p>(M^T N)^-1 = (−MN)^-1 = −(MN)^-1 = −N^-1 M^-1 (since MN = NM, matrices commute)</p><p><strong>Step 3: Simplify (MN^-1)^T</strong></p><p>(MN^-1)^T = (N^-1)^T M^T = −N^-1(−M) = N^-1 M (using skew-symmetric property and commutativity)</p><p><strong>Step 4: Compute the full expression</strong></p><p>M^2 N^2 (M^T N)^-1 (MN^-1)^T = M^2 N^2 · [−N^-1 M^-1] · [N^-1 M]</p><p>= −M^2 N^2 N^-1 M^-1 N^-1 M</p><p>= −M^2 N M^-1 N^-1 M (using N·N^-1 = I)</p><p>= −M^2 (NM^-1 N^-1) M (regrouping)</p><p><strong>Step 5: Apply commutativity MN = NM</strong></p><p>Since MN = NM, we have NM^-1 = M^-1 N, so NM^-1 N^-1 = M^-1 NN^-1 = M^-1</p><p>= −M^2 M^-1 M = −M^2</p><p>∴ Answer: C</p>
Correct Answer: C