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: Build B and C
$B=\begin{bmatrix}4&2\\2&2\end{bmatrix}$, $C=\begin{bmatrix}0&-1\\1&0\end{bmatrix}$.

Step 2: Compare the numbers
$|A|=5$, $|B|=4$, $|C|=1$. Only the sum relation holds, option (B).

Final Answer:
Because $5=4+1$, option (B) is correct. \[ \boxed{|A|=|B|+|C|} \]