Q106 / 180 · COMEDK 2012 · Mathematics

If $A=\left[\begin{array}{lll}3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1\end{array}\right], B=(\operatorname{adj} A)$ and $C=5 A$, then $\frac{|C|}{|\operatorname{adj} B|}$ is equal to

  • A
    25
  • Correct answer
    B
    125
  • C
    5
  • D
    $-5$

Solution

We have, $$ \begin{aligned} A &=\left[\begin{array}{lll} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{array}\right] \\ \text { Now, }|A| &=\left[\begin{array}{lll} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{array} \mid\right.\\ &=3(-3+4)+3(2-0)+4(-2-0) \\ &=3+6-8=1 \\ \text { Now, } \frac{|C|}{|\operatorname{adj} B|} &=\frac{|5 A|}{|B|^{3-1}}=\frac{5^{3}|A|}{|B|^{2}}=\frac{5^{3}|A|}{|\operatorname{adj} A|^{2}} \\ &=\frac{5^{3}|A|}{\left(|A|^{2}\right)^{2}}=\frac{5^{3}|A|}{|A|^{4}}=\frac{5^{3}}{|A|^{3}} \\ &=\frac{5^{3}}{1^{3}}=5^{3} \end{aligned} $$

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

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