Matrices & Determinants
Symmetric and Skew Symmetric Matrices
Grade 12
Question:
<p>If \(A\) is a symmetric matrix and \(B\) is a skew-symmetric matrix such that \(A + B = \begin{bmatrix} 2 & 3 \\ 5 & -1 \end{bmatrix}\), then \(AB\) is equal to:</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: Decompose the given matrix into symmetric and skew-symmetric parts using A = (M + M^T)/2 and B = (M - M^T)/2, where M is the given matrix. This uniquely determines both A and B.
<p><strong>Step 1:</strong> Use the unique decomposition property. For any matrix M, the symmetric part is A = (M + M^T)/2 and skew-symmetric part is B = (M - M^T)/2.</p><p><strong>Step 2:</strong> Calculate M^T: M^T = \begin{bmatrix} 2 & 5 \\ 3 & -1 \end{bmatrix}</p><p><strong>Step 3:</strong> Find A = (M + M^T)/2 = \frac{1}{2}\begin{bmatrix} 4 & 8 \\ 8 & -2 \end{bmatrix} = \begin{bmatrix} 2 & 4 \\ 4 & -1 \end{bmatrix}</p><p><strong>Step 4:</strong> Find B = (M - M^T)/2 = \frac{1}{2}\begin{bmatrix} 0 & -2 \\ 2 & 0 \end{bmatrix} = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}</p><p><strong>Step 5:</strong> Calculate AB = \begin{bmatrix} 2 & 4 \\ 4 & -1 \end{bmatrix}\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} = \begin{bmatrix} 4 & -2 \\ -1 & -4 \end{bmatrix}</p><p>∴ Answer: C</p>
Correct Answer: C