Matrices & Determinants
Symmetric and Skew-Symmetric Matrices
Grade None
Question:
<p>Let \(A + B = \begin{bmatrix} 2 & 3 \\ 5 & -1 \end{bmatrix}\) where \(A\) is a symmetric matrix and \(B\) is a skew-symmetric matrix. Find \(AB\).</p>
<p>\(\begin{bmatrix} 4 & -2 \\ -1 & -4 \end{bmatrix}\)</p>
<p>\(\begin{bmatrix} 4 & 2 \\ 1 & -4 \end{bmatrix}\)</p>
<p>\(\begin{bmatrix} -4 & 2 \\ 1 & 4 \end{bmatrix}\)</p>
<p>\(\begin{bmatrix} 4 & -2 \\ 1 & 4 \end{bmatrix}\)</p>
Step-by-Step Solution
Key Concept: Any matrix can be uniquely decomposed into symmetric and skew-symmetric parts using A = (M + M^T)/2 and B = (M - M^T)/2. Apply this decomposition to find A and B separately, then multiply them.
<p><strong>Step 1:</strong> Let M = A + B = <mmatrix>2 & 3 \\ 5 & -1</mmatrix>. Since A is symmetric (A^T = A) and B is skew-symmetric (B^T = -B), we use the standard decomposition formulas.</p><p><strong>Step 2:</strong> Calculate M^T = <mmatrix>2 & 5 \\ 3 & -1</mmatrix></p><p><strong>Step 3:</strong> Apply formulas: A = (M + M^T)/2 = ½<mmatrix>2+2 & 3+5 \\ 5+3 & -1-1</mmatrix> = ½<mmatrix>4 & 8 \\ 8 & -2</mmatrix> = <mmatrix>2 & 4 \\ 4 & -1</mmatrix></p><p><strong>Step 4:</strong> Apply formulas: B = (M - M^T)/2 = ½<mmatrix>2-2 & 3-5 \\ 5-3 & -1+1</mmatrix> = ½<mmatrix>0 & -2 \\ 2 & 0</mmatrix> = <mmatrix>0 & -1 \\ 1 & 0</mmatrix></p><p><strong>Step 5:</strong> Compute AB = <mmatrix>2 & 4 \\ 4 & -1</mmatrix><mmatrix>0 & -1 \\ 1 & 0</mmatrix> = <mmatrix>2(0)+4(1) & 2(-1)+4(0) \\ 4(0)+(-1)(1) & 4(-1)+(-1)(0)</mmatrix> = <mmatrix>4 & -2 \\ -1 & -4</mmatrix></p><p>∴ Answer: A</p>
Correct Answer: A