Find x, if `[(1,2,x),(1,1,1),(2,1,-1)]` is singualr:

Options

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

Answer

To find the value of \( x \) such that the matrix

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

is singular, we need to compute the determinant of the matrix and set it equal to zero.

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

The determinant of a 3x3 matrix

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

is given by the formula:

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

For our matrix, we have:

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

### Step 2: Calculate each term in the determinant

1. Calculate \( 1 \cdot (-1 - 1) = 1 \cdot (-2) = -2 \)
2. Calculate \( -2 \cdot (-1 - 2) = -2 \cdot (-3) = 6 \)
3. Calculate \( x \cdot (1 - 2) = x \cdot (-1) = -x \)

### Step 3: Combine the results

Now, substituting these values into the determinant expression:

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

This simplifies to:

\[ \text{det} = 4 - x \]

### Step 4: Set the determinant to zero

Since the matrix is singular, we set the determinant equal to zero:

\[ 4 - x = 0 \]

### Step 5: Solve for \( x \)

Rearranging gives:

\[ x = 4 \]

### Conclusion

Thus, the value of \( x \) for which the matrix is singular is:

\[ \boxed{4} \]