Matrices & Determinants
Symmetric and Skew-symmetric matrices
Grade 12
Question:
<p>If \(A_1, A_2, \ldots, A_{2n-1}\) are \(n\) skew-symmetric matrices of same order, then \(B = \displaystyle\sum_{r=1}^{n}(2r-1)(A_{2r-1})^{2r-1}\) will be</p>
<p>(1) symmetric</p>
<p>(2) skew-symmetric</p>
<p>(3) neither symmetric nor skew-symmetric</p>
<p>(4) data not adequate</p>
Step-by-Step Solution
Key Concept: For a skew-symmetric matrix A, we have A^T = -A. The key insight is determining the parity of matrix powers: odd powers of skew-symmetric matrices remain skew-symmetric, while even powers become symmetric. Here, (2r-1) is always odd, so each term (A_{2r-1})^{2r-1} is skew-symmetric.
<p><strong>Step 1:</strong> Recall that A is skew-symmetric if A^T = -A. For odd powers of a skew-symmetric matrix: (A^k)^T = (A^T)^k = (-A)^k = -A^k (since k is odd). Thus A^k is skew-symmetric when k is odd.</p><p><strong>Step 2:</strong> In our sum, each exponent is (2r-1), which is odd for all r = 1, 2, ..., n. Therefore each term (A_{2r-1})^{2r-1} is skew-symmetric.</p><p><strong>Step 3:</strong> The scalar coefficient (2r-1) in front of each skew-symmetric term doesn't change its nature. A scalar multiple of a skew-symmetric matrix is skew-symmetric: [(2r-1)A]^T = (2r-1)A^T = (2r-1)(-A) = -(2r-1)A.</p><p><strong>Step 4:</strong> The sum of skew-symmetric matrices is skew-symmetric: if B₁ and B₂ are skew-symmetric, then (B₁ + B₂)^T = B₁^T + B₂^T = -B₁ - B₂ = -(B₁ + B₂).</p><p><strong>Step 5:</strong> Therefore B = Σ(2r-1)(A_{2r-1})^{2r-1} is a sum of skew-symmetric matrices, making B itself skew-symmetric.</p><p>∴ Answer: B (skew-symmetric)</p>
Correct Answer: B