Q83 / 180 · COMEDK 2019 · Mathematics

If $A(\operatorname{adj} A)=5 I$, where $I$ is the identity matrix of order 3 , then $|\operatorname{adj} A|$ is equal to

  • A
    125
  • Correct answer
    B
    25
  • C
    5
  • D
    10

Solution

We have, $$ \begin{array}{ll} & A(\operatorname{adj} A)=5 I \\ \Rightarrow \quad & |A(\operatorname{adj} A)|=|5 I| \\ \Rightarrow \quad & |A||\operatorname{adj} A|=5^{3}|I| \quad\left[|K B|=K^{n}|B|\right] \\ \Rightarrow \quad & |A||A|^{3-1}=125 \times I \quad\left[|\operatorname{adj} A|=|A|^{n-1}\right] \\ \Rightarrow \quad & |A|^{3}=125 \end{array} $$ $\Rightarrow$ $$ |A|=5 $$ Now, $|\operatorname{adj} A|=|A|^{n-1}$ $$ =|A|^{2} $$ $[n=3]$ $$ =(5)^{2}=25 $$

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

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