If the matrix `[{:(1,2,x),(1,1,1),(2,1,-1):}]` is singular , then the value of x is :

Options

A. 4
B. 3
C. 2
D. 1

Answer

To find the value of \( x \) for which the matrix

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

is singular, we need to calculate the determinant of the matrix and set it 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:

- \( a = 1, b = 2, c = x \)
- \( d = 1, e = 1, f = 1 \)
- \( g = 2, h = 1, i = -1 \)

Substituting these values into the determinant formula:

\[ \text{det} = 1 \cdot (1 \cdot (-1) - 1 \cdot 1) - 2 \cdot (1 \cdot (-1) - 1 \cdot 2) + x \cdot (1 \cdot 1 - 1 \cdot 2) \]

### Step 2: Simplify the determinant expression

Calculating each term:

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

2. Second term:
\[ -2 \cdot (-1 - 2) = -2 \cdot (-3) = 6 \]

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

Putting it all together, we have:

\[ \text{det} = -2 + 6 - x = 4 - x \]

### Step 3: Set the determinant equal to zero

For the matrix to be singular, the determinant must be zero:

\[ 4 - x = 0 \]

### Step 4: Solve for \( x \)

Rearranging the equation gives:

\[ x = 4 \]

Thus, the value of \( x \) for which the matrix is singular is \( \boxed{4} \).

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