Question

Let \(S\) be a symmetric matrix obtained from \[ A= \begin{bmatrix} 1 & 2 & -3\\ 2 & -2 & 1\\ 3 & 1 & -1 \end{bmatrix} \] and \(T\) be a skew-symmetric matrix obtained from \[ B= \begin{bmatrix} 4 & 2 & 0\\ 1 & -1 & 3\\ 0 & 2 & -3 \end{bmatrix} \] If trace of \(S=-4\) and the non-zero elements of \(T\) are \(-1,1\), then \(S+T=\)

• A. \[ \begin{bmatrix} 2 & 5 & 0\\ 3 & -4 & 3\\ 0 & 1 & -2 \end{bmatrix} \]
• B. \[ \begin{bmatrix} 2 & 5 & 0\\ 3 & 3 & 3\\ 0 & 1 & 2 \end{bmatrix} \]
• C. \[ \begin{bmatrix} 2 & 5 & 0\\ 3 & -3 & 3\\ 0 & 1 & -2 \end{bmatrix} \]
• D. \[ \begin{bmatrix} 2 & -5 & 0\\ 3 & -3 & 3\\ 0 & 1 & -2 \end{bmatrix} \]

Hint:

For any square matrix \(A\): \[ \text{Symmetric part}=\frac{A+A^T}{2} \] \[ \text{Skew-symmetric part}=\frac{A-A^T}{2} \] These formulas are extremely important in matrix decomposition problems.

Answer 1

The Correct Option is A

Step 1: Know the two parts of a matrix.
Any square matrix splits into a symmetric part $\dfrac{A+A^T}{2}$ and a skew-symmetric part $\dfrac{A-A^T}{2}$. The symmetric part is mirror equal across the diagonal; the skew part has zeros on the diagonal.
Step 2: Build the symmetric matrix $S$ from $A$.
Add $A$ to its transpose and halve. This gives \[ S=\begin{bmatrix}1&2&0\\2&-2&1\\0&1&-1\end{bmatrix}. \]
Step 3: Scale $S$ to match the trace clue.
The trace of this $S$ is $1-2-1=-2$. The question wants trace $-4$, which is twice as big. So the intended $S$ is twice this, giving diagonal that fits the options.
Step 4: Build the skew matrix $T$ from $B$.
Subtract $B^T$ from $B$ and halve. The off-diagonal entries become $1$ and $-1$, matching the clue that the non-zero elements are $-1,1$.
Step 5: Add $S$ and $T$.
Adding the scaled symmetric part and the skew part position by position gives \[ S+T=\begin{bmatrix}2&5&0\\3&-4&3\\0&1&-2\end{bmatrix}. \]
Step 6: Match with the options.
This matches option 1 exactly, so that is the answer. \[ \boxed{\begin{bmatrix}2&5&0\\3&-4&3\\0&1&-2\end{bmatrix}} \]