Q73 / 180 · KCET 2014 · Mathematics
The inverse of the matrix \( A=\left[\begin{array}{lll}2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4\end{array}\right] \) is
- A\( \left[\begin{array}{lll}2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4\end{array}\right] \)
- Correct answerB\( \frac{1}{24}\left[\begin{array}{lll}1 / a & 0 & 0 \\ 0 & 1 / b & 0 \\ 0 & 0 & 1 / c\end{array}\right] \)
- C\( \frac{1}{24}\left[\begin{array}{lll}2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4\end{array}\right] \)
- D\( \frac{1}{24}\left[\begin{array}{llll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right] \)
Solution
Given that \( A=\left[\begin{array}{lll}2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4\end{array}\right] \)
Now \( A^{-1}=\left[\begin{array}{ccc}1 / 2 & 0 & 0 \\ 0 & 1 / 3 & 0 \\ 0 & 0 & 1 / 4\end{array}\right] \)
If \( A=\left[\begin{array}{ccc}a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c\end{array}\right] \), then \( A^{-1}=\left[\begin{array}{ccc}1 / a & 0 & 0 \\ 0 & 1 / b & 0 \\ 0 & 0 & 1 / c\end{array}\right] \)
When \( a \neq 0, b \neq 0, c \neq 0 \)
Now \( A^{-1}=\left[\begin{array}{ccc}1 / 2 & 0 & 0 \\ 0 & 1 / 3 & 0 \\ 0 & 0 & 1 / 4\end{array}\right] \)
If \( A=\left[\begin{array}{ccc}a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c\end{array}\right] \), then \( A^{-1}=\left[\begin{array}{ccc}1 / a & 0 & 0 \\ 0 & 1 / b & 0 \\ 0 & 0 & 1 / c\end{array}\right] \)
When \( a \neq 0, b \neq 0, c \neq 0 \)
Hence, the correct answer is option: 'B'.