Question

If \( \begin{bmatrix} 1 & -1 & x \\ 1 & x & 1 \\ x & -1 & 1 \end{bmatrix} \) has no inverse, then the real value of \( x \) is

• A. 2
• B. 3
• C. 0
• D. 1

Hint:

No inverse = Determinant is zero.

Answer 1

The Correct Option is D

Step 1: Singularity Condition
A matrix has no inverse if its determinant is zero.
Step 2: Calculation
\( \begin{vmatrix} 1 & -1 & x \\ 1 & x & 1 \\ x & -1 & 1 \end{vmatrix} = 0 \)
Step 3: Expansion
\( 1(x + 1) - (-1)(1 - x) + x(-1 - x^{2}) = 0 \) \( \Rightarrow x + 1 + 1 - x - x - x^{3} = 0 \) \( \Rightarrow -x^{3} - x + 2 = 0 \Rightarrow x^{3} + x - 2 = 0 \)
Step 4: Inspection
If we put \( x = 1 \), column 1 and column 3 become identical. \( 1^{3} + 1 - 2 = 0 \), so \( x = 1 \) is the real root.
Final Answer: (d)