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Question:

If $A=\left[\begin{array}{ll}0 & 3 \\ 2 & 0\end{array}\right]$ and $A^{-1}=\lambda(\operatorname{adj} A)$, then $\lambda=$

Solution:

Given:

$A=\left[\begin{array}{ll}0 & 3 \\ 2 & 0\end{array}\right]$

$A^{-1}=\lambda(\operatorname{adj} A)$

Now,

$A=\left[\begin{array}{ll}0 & 3 \\ 2 & 0\end{array}\right]$

$\Rightarrow|A|=\left|\begin{array}{ll}0 & 3 \\ 2 & 0\end{array}\right|$

$=-6$

As we know that,

$A^{-1}=\frac{\operatorname{adj} A}{|A|}$

$\Rightarrow \lambda(\operatorname{adj} A)=\frac{1}{|A|}(\operatorname{adj} A)$

$\Rightarrow \lambda=\frac{1}{|A|}$

$\Rightarrow \lambda=\frac{1}{-6}$

$\Rightarrow \lambda=-\frac{1}{6}$

Hence, $\lambda=-\frac{1}{6}$.