The values of $t$ such that the matrix $\left(\begin{array}{ccc}1 & 3 & 2 \\ 2 & 5 & t \\ 4 & 7-t &…

The values of $t$ such that the matrix $\left(\begin{array}{ccc}1 & 3 & 2 \\ 2 & 5 & t \\ 4 & 7-t & -6\end{array}\right)$ has no inverse, are
  1. 3, 2
  2. 3, - 2
  3. -3, 2
  4. -3, -2

Solution

Let $A=\left[\begin{array}{ccc}1 & 3 & 2 \\ 2 & 5 & t \\ 4 & 7-t & -6\end{array}\right]$ has no inverse $ \begin{aligned} & \text { So, }|A|=0 \\ & \Rightarrow 1[-30-t(7-t)]-3[-12-4 t] \\ & +2[2(7-t)-20]=0 \\ & \Rightarrow-30-7 t+t^2+36+12 t+28-4 t-40=0 \\ & \Rightarrow \quad t^2+t-6=0 \Rightarrow t^2+3 t-2 t-6=0 \\ & \Rightarrow \quad t(t+3)-2(t+3)=0 \Rightarrow(t+3)(t-2)=0 \\ & \Rightarrow \quad t=-3,+2 \\ & \text{So, values of} \quad t=-3 \text { and } 2 \\ & \end{aligned} $

Asked in: AP EAMCET 2018 (23 Apr Shift 1)

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