If `[(1,3,9),(1,x,x^(2)),(4,6,9)]` is singular matrix then x =

Options

A. A) 3
B. B) 3 or 6
C. C) 3 or `(3)/(2)`
D. D) `-3,(3)/(2)`

Answer

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

\[ \begin{pmatrix} 1 & 3 & 9 \\ 1 & x & x^2 \\ 4 & 6 & 9 \end{pmatrix} \]

is singular, we need to find when its determinant is equal to zero.

### Step 1: Calculate the Determinant
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:

\[ a = 1, b = 3, c = 9, d = 1, e = x, f = x^2, g = 4, h = 6, i = 9 \]

Substituting these values into the determinant formula:

\[ \text{det} = 1 \cdot (x \cdot 9 - x^2 \cdot 6) - 3 \cdot (1 \cdot 9 - x^2 \cdot 4) + 9 \cdot (1 \cdot 6 - x \cdot 4) \]

### Step 2: Simplify the Determinant
Now, let's simplify each term:

1. First term:
\[ 1 \cdot (9x - 6x^2) = 9x - 6x^2 \]

2. Second term:
\[ -3 \cdot (9 - 4x^2) = -27 + 12x^2 \]

3. Third term:
\[ 9 \cdot (6 - 4x) = 54 - 36x \]

Combining these, we have:

\[ \text{det} = (9x - 6x^2) + (12x^2 - 27) + (54 - 36x) \]

### Step 3: Combine Like Terms
Now, let's combine like terms:

\[ \text{det} = -6x^2 + 12x^2 + 9x - 36x + 54 - 27 \]

This simplifies to:

\[ \text{det} = 6x^2 - 27x + 27 \]

### Step 4: Set the Determinant to Zero
For the matrix to be singular, we set the determinant equal to zero:

\[ 6x^2 - 27x + 27 = 0 \]

### Step 5: Simplify the Quadratic Equation
Dividing the entire equation by 3:

\[ 2x^2 - 9x + 9 = 0 \]

### Step 6: Factor the Quadratic
Now, we will factor the quadratic:

\[ 2x^2 - 6x - 3x + 9 = 0 \]

Grouping gives us:

\[ 2x(x - 3) - 3(x - 3) = 0 \]

Factoring out \( (x - 3) \):

\[ (2x - 3)(x - 3) = 0 \]

### Step 7: Solve for \( x \)
Setting each factor to zero gives:

1. \( 2x - 3 = 0 \) implies \( x = \frac{3}{2} \)
2. \( x - 3 = 0 \) implies \( x = 3 \)

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

\[ x = 3 \quad \text{or} \quad x = \frac{3}{2} \]

### Final Answer
The values of \( x \) are \( 3 \) and \( \frac{3}{2} \).

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