Question:
If the matrix $A=\left[\begin{array}{lll}1 & a & 2 \\ 1 & 2 & 5 \\ 2 & 1 & 1\end{array}\right]$ is not invertible, than $a=$
Solution:
Given: $A=\left[\begin{array}{lll}1 & a & 2 \\ 1 & 2 & 5 \\ 2 & 1 & 1\end{array}\right]$
$A$ is not invertible if $|A|=0$.
$\left|\begin{array}{lll}1 & \alpha & 2 \\ 1 & 2 & 5 \\ 2 & 1 & 1\end{array}\right|=0$
$\Rightarrow 1(2-5)-1(\alpha-2)+2(5 \alpha-4)=0$
$\Rightarrow 1(-3)-1 \alpha+2+10 \alpha-8=0$
$\Rightarrow-3-\alpha+2+10 \alpha-8=0$
$\Rightarrow 9 \alpha-9=0$
$\Rightarrow 9 \alpha=9$
$\Rightarrow \alpha=1$
Hence, if the matrix $A=\left[\begin{array}{lll}1 & a & 2 \\ 1 & 2 & 5 \\ 2 & 1 & 1\end{array}\right]$ is not invertible, than $a=1$.