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. 1
B. 0
C. 4
D. none of these

Answer

To solve the problem, we need to find the value of \( (a + b + c)^2 \) given that the matrix

\[ A = \begin{pmatrix} 0 & a & 1 \\ -1 & b & 1 \\ -1 & c & 0 \end{pmatrix} \]

is a skew-symmetric matrix. A matrix is skew-symmetric if \( A = -A^T \).

### Step-by-step Solution:

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

2. **Set Up the Skew-Symmetric Condition**:
For \( A \) to be skew-symmetric, we have:
\[ A = -A^T \]
This gives us the equation:
\[ \begin{pmatrix} 0 & a & 1 \\ -1 & b & 1 \\ -1 & c & 0 \end{pmatrix} = -\begin{pmatrix} 0 & -1 & -1 \\ a & b & c \\ 1 & 1 & 0 \end{pmatrix} \]

3. **Equate the Matrices**:
This leads to the following equations by equating corresponding elements:
- From \( 0 = 0 \) (no new information)
- From \( a = 1 \)
- From \( 1 = -1 \) (no new information)
- From \( -1 = -a \) gives \( a = 1 \)
- From \( b = -1 \)
- From \( 1 = -c \) gives \( c = -1 \)

4. **Summing Up the Values**:
Now we have:
- \( a = 1 \)
- \( b = 0 \)
- \( c = -1 \)

Therefore, we can calculate:
\[ a + b + c = 1 + 0 - 1 = 0 \]

5. **Calculating the Square**:
Finally, we need to find \( (a + b + c)^2 \):
\[ (a + b + c)^2 = 0^2 = 0 \]

### Final Answer:
Thus, the value of \( (a + b + c)^2 \) is \( \boxed{0} \).