If $A$ is a square matrix of order 3 , then consider the following statements. I. If $|A|=0$, then $|\operatorname{Adj} A|=0$ II. If $|A|
eq 0$, then $\left|A^{-1} ight|=|A|^{-1}$ Which of the above statements is/are true?

Question

If $A$ is a square matrix of order 3 , then consider the following statements. I. If $|A|=0$, then $|\operatorname{Adj} A|=0$ II. If $|A| 
eq 0$, then $\left|A^{-1}ight|=|A|^{-1}$ Which of the above statements is/are true?

Aicharya · Accepted Answer

For a square matrix $A$ of order 3 , $ |\operatorname{Adj} \cdot A|=|A|^{3-1}=|A|^2 $ If $|A|=0$, then $|\operatorname{Adj} \cdot A|=0$ and $A \cdot A^{-1}=I$, if $|A| 
eq 0$ $ \begin{aligned} & \Rightarrow \quad\left|A \cdot A^{-1}ight|=|I| \Rightarrow|A|\left|A^{-1}ight|=1 \ & \Rightarrow \quad\left|A^{-1}ight|=|A|^{-1} \ & \end{aligned} $ So, statements I and II, both are correct

If $A$ is a square matrix of order 3 , then consider the following statements. I. If $|A|=0$, then…

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