If $\begin{bmatrix}1&3&9\\1&x&x^2\end{bmatrix}$ is singular matrix, where $x ∈ N$(where N set of natural number), then $x$ is equal to

3

The correct answer is Option (3) → 3 **

$\text{Matrix}=\begin{pmatrix}1&3&9\\1&x&x^2\\1&1&1\end{pmatrix}$

$\det=1\begin{vmatrix}x&x^2\\1&1\end{vmatrix} -3\begin{vmatrix}1&x^2\\1&1\end{vmatrix} +9\begin{vmatrix}1&x\\1&1\end{vmatrix}$

$=1(x-x^2)-3(1-x^2)+9(1-x)$

$=x-x^2-3+3x^2+9-9x$

$=2x^2-8x+6$

Singular $\Rightarrow\det=0$

$2x^2-8x+6=0$

$x^2-4x+3=0$

$(x-1)(x-3)=0$

Natural numbers give $x=1,3$

The natural number values of $x$ are $1$ and $3$.