The value of x for which the matrix `|(-x,x,2),(2,x,-x),(x,-2,-x)|` will be non-singular, are

Options

A. `-2le x le2`
B. for all x other than 2 and -2
C. `xge2`
D. `xle-2`0

Answer

To determine the values of \( x \) for which the matrix

\[ \begin{pmatrix} -x & x & 2 \\ 2 & x & -x \\ x & -2 & -x \end{pmatrix} \]

is non-singular, we need to find when the determinant of the matrix is not equal to zero.

### Step 1: Calculate the determinant of the matrix

The determinant of a \( 3 \times 3 \) matrix

\[ \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \]

is given by the formula:

\[ \text{det} = a(ei - fh) - b(di - fg) + c(dh - eg) \]

For our matrix, we have:

- \( a = -x \), \( b = x \), \( c = 2 \)
- \( d = 2 \), \( e = x \), \( f = -x \)
- \( g = x \), \( h = -2 \), \( i = -x \)

Now substituting these values into the determinant formula:

\[ \text{det} = -x \left( x(-x) - (-x)(-2) \right) - x \left( 2(-x) - (-x)(x) \right) + 2 \left( 2(-2) - x(x) \right) \]

### Step 2: Simplify the determinant expression

Calculating each term:

1. First term:
\[ -x \left( -x^2 - 2x \right) = -x \left( -x^2 - 2x \right) = x^3 + 2x^2 \]

2. Second term:
\[ -x \left( -2x + x^2 \right) = -x \left( x^2 - 2x \right) = -x^3 + 2x^2 \]

3. Third term:
\[ 2 \left( -4 - x^2 \right) = -8 - 2x^2 \]

Combining these results:

\[ \text{det} = (x^3 + 2x^2) + (-x^3 + 2x^2) + (-8 - 2x^2) \]

This simplifies to:

\[ \text{det} = 4x^2 - 8 \]

### Step 3: Set the determinant not equal to zero

To find when the matrix is non-singular, we set the determinant not equal to zero:

\[ 4x^2 - 8 \neq 0 \]

### Step 4: Solve the inequality

Solving for \( x \):

\[ 4x^2 \neq 8 \]
\[ x^2 \neq 2 \]

Taking the square root:

\[ x \neq \pm \sqrt{2} \]

### Conclusion

Thus, the values of \( x \) for which the matrix is non-singular are all real numbers except \( \sqrt{2} \) and \( -\sqrt{2} \).