Question

If the matrix \(A = [\begin{array}{cc}4 & 1 \\ 3 & 2\end{array}]\) is expressed as the sum of a symmetric matrix B and a skew symmetric matrix C then which of the following relations is correct?

• A. \(|A| = |B|\times |C|\)
• B. \(|A| = |B|+|C|\)
• C. \(|C| = 0\)
• D. \(|A| = |B|\)

Hint:

Find \(B=\frac12(A+A^T)\) and \(C=\frac12(A-A^T)\) and compare determinants.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
Every square matrix is the sum of a symmetric matrix \(B=\tfrac12(A+A^T)\) and a skew-symmetric matrix \(C=\tfrac12(A-A^T)\).

Step 2: Key Formula or Approach
For \(A=\begin{bmatrix}4&1\\3&2\end{bmatrix}\), \(A^T=\begin{bmatrix}4&3\\1&2\end{bmatrix}\).

Step 3: Detailed Explanation
\(B=\begin{bmatrix}4&2\\2&2\end{bmatrix}\) and \(C=\begin{bmatrix}0&-1\\1&0\end{bmatrix}\).
\(|A|=8-3=5\), \(|B|=8-4=4\), \(|C|=0+1=1\).
So \(|A|=|B|+|C|\) since \(5=4+1\).
The product \(4\times1=4\neq5\), and \(|C|=1\neq0\).

Final Answer:
The correct relation is \(|A|=|B|+|C|\), option (B). \[ \boxed{|A|=|B|+|C|\ \text{(B)}} \]