Matrices & Determinants
Symmetric and Skew-Symmetric Matrices
Grade None
Question:
<p>If A₁, A₃, ..., A_{2n−1} are n skew-symmetric matrices of same order, then \[X = \sum_{r=1}^{n} (2r-1)(A_{2r-1})^{2r-1}\] will be</p>
<p>(a) symmetric</p>
<p>(b) skew-symmetric</p>
<p>(c) neither symmetric nor skew symmetric</p>
<p>(d) depends on 'n' is even or odd</p>
Step-by-Step Solution
Key Concept: For any skew-symmetric matrix A, we have A^T = -A. When raising a skew-symmetric matrix to an odd power, the result is skew-symmetric, but when computing (A^T)^odd = (-A)^odd = -(A^odd), the parity of operations determines the final symmetry property.
<p><strong>Step 1:</strong> Recall that for skew-symmetric matrix A: A<sup>T</sup> = -A</p><p><strong>Step 2:</strong> For odd power k of skew-symmetric matrix A: (A<sup>k</sup>)<sup>T</sup> = (A<sup>T</sup>)<sup>k</sup> = (-A)<sup>k</sup> = -A<sup>k</sup> (since k is odd)</p><p><strong>Step 3:</strong> Each term in the sum is (2r-1)(A_{2r-1})^{2r-1} where (2r-1) is an odd coefficient and (2r-1) is an odd exponent. Taking transpose:</p><p>[(2r-1)(A_{2r-1})^{2r-1}]<sup>T</sup> = (2r-1)[(A_{2r-1})^{2r-1}]<sup>T</sup> = (2r-1)[-(A_{2r-1})^{2r-1}] = -(2r-1)(A_{2r-1})^{2r-1}</p><p><strong>Step 4:</strong> Therefore: X<sup>T</sup> = Σ[-(2r-1)(A_{2r-1})^{2r-1}] = -X</p><p><strong>Step 5:</strong> Since X<sup>T</sup> = -X, the matrix X is <strong>skew-symmetric</strong></p><p>∴ Answer: B</p>
Correct Answer: B