Find the value of $x$, if $\begin{bmatrix} 1 & x & 1 \end{bmatrix} \begin{bmatrix} 1 & 3 & 2 \\ 2 & 5 & 1 \\ 15 & 3 & 2 \end{bmatrix} \begin{bmatrix} 1 \\ 2 \\ x \end{bmatrix} = O$.

$-2, -14$

The correct answer is Option (2) → $-2, -14$ ##

We have,

$\begin{bmatrix} 1 & x & 1 \end{bmatrix}_{1 \times 3} \begin{bmatrix} 1 & 3 & 2 \\ 2 & 5 & 1 \\ 15 & 3 & 2 \end{bmatrix}_{3 \times 3} \begin{bmatrix} 1 \\ 2 \\ x \end{bmatrix}_{3 \times 1} = O$

Multiplying the first two matrices:

$\Rightarrow \begin{bmatrix} 1 + 2x + 15 & 3 + 5x + 3 & 2 + x + 2 \end{bmatrix}_{1 \times 3} \begin{bmatrix} 1 \\ 2 \\ x \end{bmatrix}_{3 \times 1} = O$

$\Rightarrow \begin{bmatrix} 16 + 2x & 5x + 6 & x + 4 \end{bmatrix}_{1 \times 3} \begin{bmatrix} 1 \\ 2 \\ x \end{bmatrix}_{3 \times 1} = O$

Now, multiplying the remaining matrices:

$\Rightarrow [16 + 2x + (5x + 6) \cdot 2 + (x + 4) \cdot x]_{1 \times 1} = O$

$\Rightarrow [16 + 2x + 10x + 12 + x^2 + 4x] = O$

$\Rightarrow [x^2 + 16x + 28] = O$

$\Rightarrow [x^2 + 2x + 14x + 28] = O$

On comparing left side with zero matrix, we get

$x^2 + 2x + 14x + 28 = 0$

$\Rightarrow (x + 2)(x + 14) = 0$

$∴x = -2, -14$