If `A={:[("0" " "a" " "1"),("-1"" "b" ""1"),("-1"" "c" ""0")]:}` is a skew-symmetric
matrix, then the value of `(a+b+c)^(2)` is:

Options

A. 0
B. 1
C. 4
D. 9

Answer

To solve the problem, we need to determine the values of \(a\), \(b\), and \(c\) in the skew-symmetric matrix \(A\) and then compute \((a + b + c)^2\).

### Step-by-step Solution:

1. **Understanding the Skew-Symmetric Matrix**:
A matrix \(A\) is skew-symmetric if \(A^T = -A\), where \(A^T\) is the transpose of matrix \(A\).

2. **Write Down the Given Matrix**:
The given skew-symmetric matrix is:
\[ A = \begin{pmatrix} 0 & a & 1 \\ -1 & b & 1 \\ -1 & c & 0 \end{pmatrix} \]

3. **Find the Transpose of the Matrix**:
The transpose \(A^T\) of matrix \(A\) is:
\[ A^T = \begin{pmatrix} 0 & -1 & -1 \\ a & b & c \\ 1 & 1 & 0 \end{pmatrix} \]

4. **Set Up the Equation for Skew-Symmetry**:
According to the definition of skew-symmetry, we have:
\[ A^T = -A \]
Therefore,
\[ \begin{pmatrix} 0 & -1 & -1 \\ a & b & c \\ 1 & 1 & 0 \end{pmatrix} = \begin{pmatrix} 0 & -a & -1 \\ 1 & -b & -1 \\ 1 & 1 & 0 \end{pmatrix} \]

5. **Equate Corresponding Elements**:
From the above matrices, we can equate corresponding elements:
- From the first row: \(0 = 0\) (no information)
- From the second row:
- \(a = -1\)
- \(b = -b\) implies \(2b = 0\) so \(b = 0\)
- \(c = -1\)
- From the third row: \(1 = 1\) (no new information)

6. **Summarize the Values**:
We have found:
- \(a = -1\)
- \(b = 0\)
- \(c = -1\)

7. **Calculate \(a + b + c\)**:
Now, we compute:
\[ a + b + c = -1 + 0 - 1 = -2 \]

8. **Calculate \((a + b + c)^2\)**:
Finally, we find:
\[ (a + b + c)^2 = (-2)^2 = 4 \]

### Final Answer:
The value of \((a + b + c)^2\) is \(4\).

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