Given ,
Nullity of a matrix = Total number columns – Rank of that matrix
Rank of that matrix =Total number columns – Nullity of a matrix
= 3 -1 = 2
We will solve this by converting the given matrix into Echelon form using Gauss Elimination method.
1)
$\Rightarrow$ $\begin{bmatrix} 2 &3 &7 \\ 1 &1 &9 \\ 9 & 2 & x \end{bmatrix}$
R3 $\leftarrow$ R3 – R2
$\Rightarrow$ $\begin{bmatrix} 2 &3 &7 \\ 1 &1 &9 \\ 8 & 1 & x-9 \end{bmatrix}$
2)
R2 $\leftarrow$ 2R2 – R1
R3 $\leftarrow$ R3 – 4R1
$\Rightarrow$ $\begin{bmatrix} 2 &3 &7 \\ 0 &-1 &9 \\ 0 & -11 & x-37 \end{bmatrix}$
3)
R3 $\leftarrow$ R3 – 11R2
$\Rightarrow$ $\begin{bmatrix} 2 &3 &7 \\ 0 &-1 &9 \\ 0 & 0 & x-37-121\end{bmatrix}$
Since there should be only two pivot column ( since the rank is 2)
therefore,
x-37-121 = 0
$\therefore$ x = 158