Q69 / 180 · COMEDK 2025 (10 May Shift 2) · Mathematics
If $A=\left[\begin{array}{ccc}0 & -1 & 2 \\ 1 & 0 & 3 \\ -2 & -3 & 0\end{array}\right]$, then $A+2 A^T=$
- A$2 A^2$
- B$A$
- C$-A^T$
- Correct answerD$A^T$
Solution
Given the matrix $A = \begin{bmatrix} 0 & -1 & 2 \\ 1 & 0 & 3 \\ -2 & -3 & 0 \end{bmatrix}$.
The transpose of matrix $A$ is $A^T = \begin{bmatrix} 0 & 1 & -2 \\ -1 & 0 & -3 \\ 2 & 3 & 0 \end{bmatrix}$.
Observe that $A^T = -A$, which implies that $A$ is a skew-symmetric matrix.
We need to compute $A + 2A^T$.
Substituting $A^T = -A$ into the expression, we get $A + 2(-A) = A - 2A = -A$.
Since $A^T = -A$, it follows that $-A = A^T$.
Therefore, $A + 2A^T = A^T$.
Answer: $A^T$
The transpose of matrix $A$ is $A^T = \begin{bmatrix} 0 & 1 & -2 \\ -1 & 0 & -3 \\ 2 & 3 & 0 \end{bmatrix}$.
Observe that $A^T = -A$, which implies that $A$ is a skew-symmetric matrix.
We need to compute $A + 2A^T$.
Substituting $A^T = -A$ into the expression, we get $A + 2(-A) = A - 2A = -A$.
Since $A^T = -A$, it follows that $-A = A^T$.
Therefore, $A + 2A^T = A^T$.
Answer: $A^T$
Hence, the correct answer is option: 'D'.