For what value of x, the matrix ` [{:( 3-x,2,2),( 2,4-x,1),(-2,-4,-1-x) :}]` is singular

Options

A. ` x=1`
B. ` x=2`
C. ` x=0`
D. ` x=3`

Answer

To determine the value 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 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:

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

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 expression

Calculating each part step by step:

1. Calculate \( (4 - x)(-1 - x) - (-4) \):

\[ (4 - x)(-1 - x) = -4 + 4x + x - x^2 = -x^2 + 5x - 4 \]

So,

\[ -x^2 + 5x - 4 + 4 = -x^2 + 5x \]

2. Calculate \( 2(-1 - x) - (-2) \):

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

3. Calculate \( 2(-4) - (4 - x)(-2) \):

\[ -8 + 2(4 - x) = -8 + 8 - 2x = -2x \]

Now substituting these back into the determinant expression:

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

This simplifies to:

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

Thus:

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

### Step 3: Set the determinant to zero

To find the values of \( x \) for which the matrix is singular, we set the determinant to zero:

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

This gives us two equations to solve:

1. \( 3 - x = 0 \) which gives \( x = 3 \)
2. \( -x^2 + 5x = 0 \) which can be factored as \( x(-x + 5) = 0 \), giving \( x = 0 \) or \( x = 5 \)

### Conclusion

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

\[ x = 0, \quad x = 3, \quad x = 5 \]