If $A=\left[\begin{array}{ccc}1 & -1 & 1 \\ 2 & -1 & 0 \\ 3 & 3 & -4\end{array}\right]$, adj…

If $A=\left[\begin{array}{ccc}1 & -1 & 1 \\ 2 & -1 & 0 \\ 3 & 3 & -4\end{array}\right]$, adj $A=\left[\begin{array}{ccc}4 & -1 & 1 \\ 8 & -7 & a \\ 9 & -6 & b\end{array}\right]$, then
  1. $a=2, b=-1$
  2. $a=2, b=1$
  3. $a=-2, b=1$
  4. $a=1, b=-2$

Solution

$\begin{aligned} & A=\left[\begin{array}{ccc}1 & -1 & 1 \\ 2 & -1 & 0 \\ 3 & 3 & -4\end{array}\right], \operatorname{adj} A=\left[\begin{array}{lll}4 & -1 & 1 \\ 8 & -7 & a \\ 9 & -6 & b\end{array}\right] \\ & \text { now, } A \cdot \operatorname{adj} A=\left[\begin{array}{ccc}5 & 0 & 1-a+b \\ 0 & 5 & 2-a \\ 0 & 0 & 3+3 a-4 b\end{array}\right]=|A| . \mathrm{I} \\ & \Rightarrow 1-a+b=0 \text { and } 2-a=0 \\ & \Rightarrow a=2 \text { and } b=1\end{aligned}$

Asked in: MHT CET 2022 (07 Aug Shift 2)

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