If `[(1,x,1)][(1,3,2),(0,5,1),(0,3,2)][(1),(1),(x)]=O` then x equals to

Options

A. `-3+-sqrt(3)`
B. `(-9+-sqrt(53))/2`
C. `1`
D. None of these

Answer

To solve the equation given by the matrix multiplication, we can break it down into a series of steps:

### Step 1: Define the Matrices
Let:
- Matrix A = \(\begin{bmatrix} 1 & x & 1 \end{bmatrix}\)
- Matrix B = \(\begin{bmatrix} 1 & 3 & 2 \\ 0 & 5 & 1 \\ 0 & 3 & 2 \end{bmatrix}\)
- Matrix C = \(\begin{bmatrix} 1 \\ 1 \\ x \end{bmatrix}\)

### Step 2: Multiply A and B
To find the product \(AB\):
\[ AB = \begin{bmatrix} 1 & x & 1 \end{bmatrix} \begin{bmatrix} 1 & 3 & 2 \\ 0 & 5 & 1 \\ 0 & 3 & 2 \end{bmatrix} \]

Calculating the elements of the resulting matrix:
- First element: \(1 \cdot 1 + x \cdot 0 + 1 \cdot 0 = 1\)
- Second element: \(1 \cdot 3 + x \cdot 5 + 1 \cdot 3 = 3 + 5x + 3 = 6 + 5x\)
- Third element: \(1 \cdot 2 + x \cdot 1 + 1 \cdot 2 = 2 + x + 2 = 4 + x\)

Thus, we have:
\[ AB = \begin{bmatrix} 1 & 6 + 5x & 4 + x \end{bmatrix} \]

### Step 3: Multiply (AB) and C
Now we multiply \(AB\) with \(C\):
\[ (AB)C = \begin{bmatrix} 1 & 6 + 5x & 4 + x \end{bmatrix} \begin{bmatrix} 1 \\ 1 \\ x \end{bmatrix} \]

Calculating the elements:
- First element: \(1 \cdot 1 + (6 + 5x) \cdot 1 + (4 + x) \cdot x\)
\[ = 1 + 6 + 5x + 4x + x^2 = x^2 + 9x + 7 \]

Thus, we have:
\[ (AB)C = \begin{bmatrix} x^2 + 9x + 7 \end{bmatrix} \]

### Step 4: Set the Result Equal to Zero
According to the problem, we set this equal to the zero matrix:
\[ x^2 + 9x + 7 = 0 \]

### Step 5: Solve the Quadratic Equation
To solve the quadratic equation \(x^2 + 9x + 7 = 0\), we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \(a = 1\), \(b = 9\), and \(c = 7\).

Calculating the discriminant:
\[ b^2 - 4ac = 9^2 - 4 \cdot 1 \cdot 7 = 81 - 28 = 53 \]

Now substituting into the quadratic formula:
\[ x = \frac{-9 \pm \sqrt{53}}{2} \]

### Conclusion
The values of \(x\) are:
\[ x = \frac{-9 + \sqrt{53}}{2} \quad \text{and} \quad x = \frac{-9 - \sqrt{53}}{2} \]