If `[(1,-1,x),(1,x,1),(x,-1,1)]` has no inverse, then the real value of `x` is

Options

A. 2
B. 3
C. 0
D. 1

Explanation

To determine the real value of \( x \) such that the matrix

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

has no inverse, we need to find when the determinant of the matrix is equal to zero. A matrix has no inverse if it is singular, which occurs when its determinant is zero.

### Step 1: Calculate the Determinant

The determinant of a 3x3 matrix

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

is given by the formula:

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

For our matrix \( A \):

- \( a = 1, b = -1, c = x \)
- \( d = 1, e = x, f = 1 \)
- \( g = x, h = -1, i = 1 \)

Substituting these values into the determinant formula:

\[ \text{det}(A) = 1 \cdot (x \cdot 1 - 1 \cdot (-1)) - (-1) \cdot (1 \cdot 1 - 1 \cdot x) + x \cdot (1 \cdot (-1) - x \cdot x) \]

### Step 2: Simplify the Determinant

Calculating each term:

1. First term:
\[ 1 \cdot (x + 1) = x + 1 \]

2. Second term:
\[ -(-1) \cdot (1 - x) = 1 - x \]

3. Third term:
\[ x \cdot (-1 - x^2) = -x - x^3 \]

Combining these results, we have:

\[ \text{det}(A) = (x + 1) + (1 - x) + (-x - x^3) \]

### Step 3: Combine Like Terms

Now, combine the terms:

\[ \text{det}(A) = x + 1 + 1 - x - x - x^3 = 2 - x - x^3 \]

### Step 4: Set the Determinant to Zero

For the matrix to have no inverse, we set the determinant to zero:

\[ 2 - x - x^3 = 0 \]

Rearranging gives:

\[ x^3 + x - 2 = 0 \]

### Step 5: Solve the Polynomial Equation

To find the real value of \( x \), we can use the Rational Root Theorem or trial and error to find possible rational roots. Testing \( x = 1 \):

\[ 1^3 + 1 - 2 = 1 + 1 - 2 = 0 \]

Thus, \( x = 1 \) is a root.

### Step 6: Factor the Polynomial

Now we can factor \( x^3 + x - 2 \) using \( (x - 1) \):

Using synthetic division or polynomial long division, we find:

\[ x^3 + x - 2 = (x - 1)(x^2 + x + 2) \]

### Step 7: Analyze the Quadratic Factor

The quadratic \( x^2 + x + 2 \) has no real roots (discriminant \( b^2 - 4ac = 1 - 8 < 0 \)). Therefore, the only real solution is:

\[ x = 1 \]

### Conclusion

Thus, the real value of \( x \) such that the matrix has no inverse is:

\[ \boxed{1} \]