Question

Let $A_t=\left[\begin{array}{ccc}1& 3& 2\\ 2& 5& t\\ 4& 7-t& -6\end{array}\right]$, then the value(s) of $t$ for which inverse of $A$, does not exist.

• A. -2, 1
• B. 3, 2
• C. 2, -3
• D. 3, -1

Answer 1

Answer: (C)

2, -3

$A_t=\left[\begin{array}{ccc}1& 3& 2\\ 2& 5& t\\ 4& 7-t& -6\end{array}\right]$
Inverse does not exist.
$\Rightarrow |A_t|=0$
$\Rightarrow \left|\begin{array}{ccc}1& 3& 2\\ 2& 5& t\\ 4& 7-t& -6\end{array}\right|=0$
$\Rightarrow (-30-7t+t^2)-3(-12-4t)+2(-2t-6)=0$
$\Rightarrow t^2+t-6=(t+3)(t-2)=0$
$\therefore t=-3$ or $t=2$
Hence, option C.