Q64 / 150 · MHT CET 2026 (17 April Shift 1) · Mathematics
If the matrix $A = \begin{bmatrix} 4 & 1 \\ 3 & 2 \end{bmatrix}$ 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$|A| = |B| \times |C|$
- Correct answerB$|A| = |B| + |C|$
- C$|C| = 0$
- D$|A| = |B|$
Solution
Given $A = \begin{bmatrix} 4 & 1 \\ 3 & 2 \end{bmatrix}$
The transpose of matrix $A$ is $A^T = \begin{bmatrix} 4 & 3 \\ 1 & 2 \end{bmatrix}$
Any square matrix can be expressed as the sum of a symmetric matrix $B$ and a skew-symmetric matrix $C$, where $B = \dfrac{1}{2}(A + A^T)$ and $C = \dfrac{1}{2}(A - A^T)$
$B = \dfrac{1}{2} \left( \begin{bmatrix} 4 & 1 \\ 3 & 2 \end{bmatrix} + \begin{bmatrix} 4 & 3 \\ 1 & 2 \end{bmatrix} \right) = \begin{bmatrix} 4 & 2 \\ 2 & 2 \end{bmatrix}$
$C = \dfrac{1}{2} \left( \begin{bmatrix} 4 & 1 \\ 3 & 2 \end{bmatrix} - \begin{bmatrix} 4 & 3 \\ 1 & 2 \end{bmatrix} \right) = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$
Calculating the determinants of $A$, $B$, and $C$:
$|A| = (4)(2) - (1)(3) = 5$
$|B| = (4)(2) - (2)(2) = 4$
$|C| = (0)(0) - (-1)(1) = 1$
Checking the given relations, we find that $|B| + |C| = 4 + 1 = 5$
Therefore, $|A| = |B| + |C|$
Answer: $|A| = |B| + |C|$
The transpose of matrix $A$ is $A^T = \begin{bmatrix} 4 & 3 \\ 1 & 2 \end{bmatrix}$
Any square matrix can be expressed as the sum of a symmetric matrix $B$ and a skew-symmetric matrix $C$, where $B = \dfrac{1}{2}(A + A^T)$ and $C = \dfrac{1}{2}(A - A^T)$
$B = \dfrac{1}{2} \left( \begin{bmatrix} 4 & 1 \\ 3 & 2 \end{bmatrix} + \begin{bmatrix} 4 & 3 \\ 1 & 2 \end{bmatrix} \right) = \begin{bmatrix} 4 & 2 \\ 2 & 2 \end{bmatrix}$
$C = \dfrac{1}{2} \left( \begin{bmatrix} 4 & 1 \\ 3 & 2 \end{bmatrix} - \begin{bmatrix} 4 & 3 \\ 1 & 2 \end{bmatrix} \right) = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$
Calculating the determinants of $A$, $B$, and $C$:
$|A| = (4)(2) - (1)(3) = 5$
$|B| = (4)(2) - (2)(2) = 4$
$|C| = (0)(0) - (-1)(1) = 1$
Checking the given relations, we find that $|B| + |C| = 4 + 1 = 5$
Therefore, $|A| = |B| + |C|$
Answer: $|A| = |B| + |C|$
Hence, the correct answer is option: 'B'.