Matrices & Determinants
Symmetric and Skew-Symmetric Matrices
Grade 12
Question:
<p>Let \(X\) and \(Y\) be two arbitrary, \(3 \times 3\), non-zero, skew-symmetric matrices and \(Z\) be an arbitrary \(3 \times 3\), non-zero, symmetric matrix. Then which of the following matrices is (are) skew symmetric?</p>
<p>\(Y^3 Z^4 - Z^4 Y^3\)</p>
<p>\(X^{44} + Y^{44}\)</p>
<p>\(X^4 Z^3 - Z^3 X^4\)</p>
<p>\(X^{23} + Y^{23}\)</p>
Step-by-Step Solution
Key Concept: A matrix is skew-symmetric if A^T = -A. Use the property that (AB)^T = B^T A^T and (A+B)^T = A^T + B^T to test each combination. For products of skew-symmetric matrices: if X^T = -X and Y^T = -Y, then (XY)^T = Y^T X^T = (-Y)(-X) = YX, which equals -XY only if XY = -YX (not generally true). However, XYX has the property (XYX)^T = X^T Y^T X^T = (-X)(-Y)(-X) = -XYX.
<p><strong>Key Property:</strong> Matrix A is skew-symmetric ⟺ A^T = -A</p><p><strong>For X, Y skew-symmetric:</strong> X^T = -X, Y^T = -Y</p><p><strong>For Z symmetric:</strong> Z^T = Z</p><p><strong>Testing XYX:</strong></p><p>(XYX)^T = X^T Y^T X^T = (-X)(-Y)(-X) = -XYX ✓ SKEW-SYMMETRIC</p><p><strong>Testing XYZ:</strong></p><p>(XYZ)^T = Z^T Y^T X^T = Z(-Y)(-X) = ZYX ≠ -XYZ (generally)</p><p><strong>Testing ZXZ:</strong></p><p>(ZXZ)^T = Z^T X^T Z^T = Z(-X)Z = -ZXZ ✓ SKEW-SYMMETRIC</p><p><strong>Testing XZY:</strong></p><p>(XZY)^T = Y^T Z^T X^T = (-Y)Z(-X) = YZX ≠ -XZY (generally)</p><p>∴ Answer: C (XYX) and D (ZXZ)</p>
Correct Answer: CD