Question

Express \( \begin{bmatrix} 5 & 1 \\ 3 & 7 \end{bmatrix} \) as the sum of a symmetric matrix and a skew-symmetric matrix.

Hint:

A quick check: The skew-symmetric matrix \( Q \) must have zeros on its main diagonal, and its elements must satisfy \( q_{ij} = -q_{ji} \). Only option (A) and (D) follow this, and \( 2 + (-1) = 1 \) (matching element \( a_{12} \) of A).

Answer 1

Step 1: Understanding the Concept:
Any square matrix \( A \) can be expressed as \( A = P + Q \), where \( P = \frac{1}{2}(A + A^T) \) is symmetric and \( Q = \frac{1}{2}(A - A^T) \) is skew-symmetric.
Step 2: Key Formula or Approach:
Find \( A^T \) (transpose), then compute \( P \) and \( Q \).
Step 3: Detailed Explanation:
Given \( A = \begin{bmatrix} 5 & 1 \\ 3 & 7 \end{bmatrix} \), then \( A^T = \begin{bmatrix} 5 & 3 \\ 1 & 7 \end{bmatrix} \).
Calculate \( P = \frac{1}{2}(A + A^T) \):
\[ P = \frac{1}{2} \left( \begin{bmatrix} 5 & 1 \\ 3 & 7 \end{bmatrix} + \begin{bmatrix} 5 & 3 \\ 1 & 7 \end{bmatrix} \right) = \frac{1}{2} \begin{bmatrix} 10 & 4 \\ 4 & 14 \end{bmatrix} = \begin{bmatrix} 5 & 2 \\ 2 & 7 \end{bmatrix} \] Calculate \( Q = \frac{1}{2}(A - A^T) \):
\[ Q = \frac{1}{2} \left( \begin{bmatrix} 5 & 1 \\ 3 & 7 \end{bmatrix} - \begin{bmatrix} 5 & 3 \\ 1 & 7 \end{bmatrix} \right) = \frac{1}{2} \begin{bmatrix} 0 & -2 \\ 2 & 0 \end{bmatrix} = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \] Thus, \( A = \begin{bmatrix} 5 & 2 \\ 2 & 7 \end{bmatrix} + \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \).
Step 4: Final Answer:
The matrix expressed as a sum is \( \begin{bmatrix} 5 & 2 \\ 2 & 7 \end{bmatrix} + \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \).