For what value of 'x', is the matrix , `A=[(0,1,-2),(-1,0,3),(x,-3,0)]` a skew - symmetric matrix

Answer

To determine the value of 'x' for which the matrix

\[ A = \begin{pmatrix} 0 & 1 & -2 \\ -1 & 0 & 3 \\ x & -3 & 0 \end{pmatrix} \]

is a skew-symmetric matrix, we need to use the property of skew-symmetric matrices. A matrix \( A \) is skew-symmetric if

\[ A^T = -A \]

where \( A^T \) is the transpose of \( A \).

### Step 1: Find the transpose of matrix A

The transpose of a matrix is obtained by swapping its rows and columns. Thus, the transpose \( A^T \) of the matrix \( A \) is:

\[ A^T = \begin{pmatrix} 0 & -1 & x \\ 1 & 0 & -3 \\ -2 & 3 & 0 \end{pmatrix} \]

### Step 2: Find -A

Next, we find \(-A\):

\[ -A = \begin{pmatrix} 0 & -1 & 2 \\ 1 & 0 & -3 \\ -x & 3 & 0 \end{pmatrix} \]

### Step 3: Set \( A^T = -A \)

For \( A \) to be skew-symmetric, we set the transpose equal to the negative of the original matrix:

\[ \begin{pmatrix} 0 & -1 & x \\ 1 & 0 & -3 \\ -2 & 3 & 0 \end{pmatrix} = \begin{pmatrix} 0 & -1 & 2 \\ 1 & 0 & -3 \\ -x & 3 & 0 \end{pmatrix} \]

### Step 4: Compare corresponding elements

Now, we compare the corresponding elements of the two matrices:

1. From the first row, first column: \( 0 = 0 \) (True)
2. From the first row, second column: \( -1 = -1 \) (True)
3. From the first row, third column: \( x = 2 \)
4. From the second row, first column: \( 1 = 1 \) (True)
5. From the second row, second column: \( 0 = 0 \) (True)
6. From the second row, third column: \( -3 = -3 \) (True)
7. From the third row, first column: \( -2 = -x \) implies \( x = 2 \)
8. From the third row, second column: \( 3 = 3 \) (True)
9. From the third row, third column: \( 0 = 0 \) (True)

### Step 5: Solve for x

From the comparisons, we find that:

\[ x = 2 \]

### Conclusion

Thus, the value of \( x \) for which the matrix \( A \) is skew-symmetric is:

\[ \boxed{2} \]

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