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$(P):$ Given that $M=\begin{bmatrix} 1&1 &1 \\ 1&2 &4 \\1 &3 &9 \end{bmatrix}$

                           $\Rightarrow$ If matrix $M$ is singular then $|M|=0$

              $|M|=\begin{vmatrix} 1&1 &1 \\ 1&2 &4 \\1 &3 &9 \end{vmatrix}$

                 $R_{3}\rightarrow R_{3}-R_{2}$

     $|M|=\begin{vmatrix} 1&1 &1 \\ 1&2 &4 \\0 &1 &5 \end{vmatrix}$

                $R_{2}\rightarrow R_{2}-R_{1}$

 $|M|=\begin{vmatrix} 1&1 &1 \\ 0&1 &3 \\0 &1 &5 \end{vmatrix}$

            $R_{3}\rightarrow R_{3}-R_{2}$

$|M|=\begin{vmatrix} 1&1 &1 \\ 0&1 &3 \\0 &0 &2 \end{vmatrix}$

     $|M|=2$

So, it is not a singular matrix.

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