If A = `[(1,0,2),(5,1,x),(1,1,1)]` is a singular matrix then x is equal to

Options

A. 3
B. 5
C. 9
D. 11

Answer

To find the value of \( x \) such that the matrix \( A = \begin{pmatrix} 1 & 0 & 2 \\ 5 & 1 & x \\ 1 & 1 & 1 \end{pmatrix} \) is singular, we need to calculate the determinant of the matrix and set it equal to zero.

### Step 1: Calculate the determinant of matrix \( A \)

The determinant of a 3x3 matrix \( A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \) is given by the formula:

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

For our matrix \( A \):

- \( a = 1, b = 0, c = 2 \)
- \( d = 5, e = 1, f = x \)
- \( g = 1, h = 1, i = 1 \)

Substituting these values into the determinant formula, we get:

\[ \text{det}(A) = 1(1 \cdot 1 - x \cdot 1) - 0(5 \cdot 1 - x \cdot 1) + 2(5 \cdot 1 - 1 \cdot 1) \]

### Step 2: Simplify the determinant expression

Calculating each term:

1. The first term:
\[ 1(1 - x) = 1 - x \]

2. The second term is zero since it is multiplied by \( b = 0 \).

3. The third term:
\[ 2(5 - 1) = 2 \cdot 4 = 8 \]

Putting it all together, we have:

\[ \text{det}(A) = (1 - x) + 8 = 9 - x \]

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

Since \( A \) is a singular matrix, we set the determinant to zero:

\[ 9 - x = 0 \]

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

Rearranging the equation gives:

\[ x = 9 \]

### Conclusion

Thus, the value of \( x \) such that the matrix \( A \) is singular is \( \boxed{9} \).

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