Q45 / 160 · AP EAMCET 2018 (22 Apr Shift 2) · Mathematics

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?

  • Correct answer
    A
    Both I and II
  • B
    Neither I nor II
  • C
    I only
  • D
    II only

Solution

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.

Hence, the correct answer is option: 'A'.

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