The number of values of x for which the matrix `A=[(3-x,2,2),(2,4-x,1),(-2,-4,-1-x)]` is singular, is

Options

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

Answer

To determine the number of values of \( x \) for which the matrix

\[ A = \begin{pmatrix} 3 - x & 2 & 2 \\ 2 & 4 - x & 1 \\ -2 & -4 & -1 - x \end{pmatrix} \]

is singular, we need to find the determinant of the matrix \( A \) and set it equal to zero. A matrix is singular if its determinant is zero.

### Step 1: Calculate the Determinant of Matrix A

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 \( A \):

- \( a = 3 - x \)
- \( b = 2 \)
- \( c = 2 \)
- \( d = 2 \)
- \( e = 4 - x \)
- \( f = 1 \)
- \( g = -2 \)
- \( h = -4 \)
- \( i = -1 - x \)

Now, substituting these values into the determinant formula:

\[ \text{det}(A) = (3 - x)((4 - x)(-1 - x) - (1)(-4)) - (2)(2(-1 - x) - (1)(-2)) + (2)(2(-4) - (4 - x)(-2)) \]

### Step 2: Simplify the Determinant

Calculating each term:

1. **First term**:
\[ (4 - x)(-1 - x) - (1)(-4) = (-4 + 4x + x + x^2) + 4 = x^2 + 5x \]
Thus,
\[ (3 - x)(x^2 + 5x) \]

2. **Second term**:
\[ 2(-1 - x) - (-2) = -2 - 2x + 2 = -2x \]
Thus,
\[ -2(-2x) = 4x \]

3. **Third term**:
\[ 2(-4) - (4 - x)(-2) = -8 + 8 - 2x = -2x \]
Thus,
\[ 2(-2x) = -4x \]

Combining all these:

\[ \text{det}(A) = (3 - x)(x^2 + 5x) + 4x - 4x = (3 - x)(x^2 + 5x) \]

### Step 3: Set the Determinant Equal to Zero

Now we need to set the determinant equal to zero:

\[ (3 - x)(x^2 + 5x) = 0 \]

This gives us two factors:

1. \( 3 - x = 0 \) → \( x = 3 \)
2. \( x^2 + 5x = 0 \) → \( x(x + 5) = 0 \) → \( x = 0 \) or \( x = -5 \)

### Step 4: List All Values of x

The values of \( x \) for which the matrix \( A \) is singular are:

- \( x = 3 \)
- \( x = 0 \)
- \( x = -5 \)

Thus, there are **3 values of \( x \)** for which the matrix is singular.

### Final Answer

The number of values of \( x \) for which the matrix \( A \) is singular is **3**.

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