Question

If the matrix A = $\left[\begin{array}{ccc}1& a& 2\\ 1& 2& 5\\ 2& 1& 1\end{array}\right]$ is not invertible, than a = ___________________.

Answer 1

Given: A = $\left[\begin{array}{ccc}1& a& 2\\ 1& 2& 5\\ 2& 1& 1\end{array}\right]$

A is not invertible if |A| = 0.
$$\begin{gathered} \left|\begin{array}{ccc}1& \alpha & 2\\ 1& 2& 5\\ 2& 1& 1\end{array}\right|=0 \\ \Rightarrow 1\left(2-5\right)-1\left(\alpha -2\right)+2\left(5\alpha -4\right)=0 \\ \Rightarrow 1\left(-3\right)-1\alpha +2+10\alpha -8=0 \\ \Rightarrow -3-\alpha +2+10\alpha -8=0 \\ \Rightarrow 9\alpha -9=0 \\ \Rightarrow 9\alpha =9 \\ \Rightarrow \alpha =1 \end{gathered}$$


Hence, if the matrix A = $\left[\begin{array}{ccc}1& a& 2\\ 1& 2& 5\\ 2& 1& 1\end{array}\right]$ is not invertible, than a = 1.