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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 $|A| \neq 0$, then $\left|A^{-1}\right|=|A|^{-1}$
Which of the above statements is/are true?
Options:
I. If $|A|=0$, then $|\operatorname{Adj} A|=0$
II. If $|A| \neq 0$, then $\left|A^{-1}\right|=|A|^{-1}$
Which of the above statements is/are true?
Solution:
1131 Upvotes
Verified Answer
The correct answer is:
Both I and II
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 $|A| \neq 0$
$$
\begin{aligned}
& \Rightarrow \quad\left|A \cdot A^{-1}\right|=|I| \Rightarrow|A|\left|A^{-1}\right|=1 \\
& \Rightarrow \quad\left|A^{-1}\right|=|A|^{-1} \\
&
\end{aligned}
$$
So, statements I and II, both are correct.
$$
|\operatorname{Adj} \cdot A|=|A|^{3-1}=|A|^2
$$
If $|A|=0$, then $|\operatorname{Adj} \cdot A|=0$
and $A \cdot A^{-1}=I$, if $|A| \neq 0$
$$
\begin{aligned}
& \Rightarrow \quad\left|A \cdot A^{-1}\right|=|I| \Rightarrow|A|\left|A^{-1}\right|=1 \\
& \Rightarrow \quad\left|A^{-1}\right|=|A|^{-1} \\
&
\end{aligned}
$$
So, statements I and II, both are correct.
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