Q108 / 180 · COMEDK 2012 · Mathematics

If $A=\left[\begin{array}{cc}1 & \log _{b} a \\ \log _{a} b & 1\end{array}\right]$, then $|A|$ is equal to

  • Correct answer
    A
    0
  • B
    $\log _{a} b$
  • C
    $-1$
  • D
    $\log _{b} a$

Solution

We have, $A=\left[\begin{array}{cc}1 & \log _{b} a \\ \log _{a} b & 1\end{array}\right]$ Now, $$ \begin{aligned} |A| &=\left|\begin{array}{cc} 1 & \log _{b} a \\ \log _{a} b & 1 \end{array}\right| \\ &=1-\log _{a} b \log _{b} a \\ &=1-\log _{a} b \times \frac{1}{\log _{a} b}\left\{\because \log _{b} a=\frac{1}{\log _{a} b}\right\} \\ &=1-1=0 \end{aligned} $$

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

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