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NDA 2017
Exam
NDA
Year
2017
Subject
Mathematics
Chapter
Matrices

NDA 2017 · Mathematics · Matrices

If a, b, c are non-zero real numbers, then the inverse of the...

If a,b,ca, b, c are non-zero real numbers, then the inverse of the \operatorname{matrix} A=\left[a000b000c\begin{array}{lll}\mathrm{a} & 0 & 0 \\ 0 & \mathrm{ b} & 0 \\ 0 & 0 & \mathrm{c}\end{array}$\right] is equal to

Options

  1. A\left[\mathrma−1000 b−1000\mathrmc−1\begin{array}{lll}\mathrma^{-1} & 0 & 0 \\ 0 & \mathrm{~b}^{-1} & 0 \\ 0 & 0 & \mathrmc^{-1}\end{array}\right]
  2. B\frac{1}{\mathrm{abc}}\left[\mathrma−1000b−1000\mathrmc−1\begin{array}{lll}\mathrma^{-1} & 0 & 0 \\ 0 & \mathrm{ b}^{-1} & 0 \\ 0 & 0 & \mathrmc^{-1}\end{array}\right]
  3. C\frac{1}{\mathrm{abc}}\left[100010001\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]
  4. D\frac{1}{\mathrm{abc}}\left[a000b000c\begin{array}{lll}\mathrm{a} & 0 & 0 \\ 0 & \mathrm{ b} & 0 \\ 0 & 0 & \mathrm{c}\end{array}\right]

Correct answer

A. \left[\mathrma−1000 b−1000\mathrmc−1\begin{array}{lll}\mathrma^{-1} & 0 & 0 \\ 0 & \mathrm{~b}^{-1} & 0 \\ 0 & 0 & \mathrmc^{-1}\end{array}\right]

Step-by-step solution

\mathrm{A}=\left[a000b000c\begin{array}{lll}\mathrm{a} & 0 & 0 \\ 0 & \mathrm{ b} & 0 \\ 0 & 0 & \mathrm{c}\end{array}\right]\mathrm{A}^{-1}=\left[1a0001b0001c\begin{array}{ccc}\frac{1}{\mathrm{a}} & 0 & 0 \\ 0 & \frac{1}{\mathrm{ b}} & 0 \\ 0 & 0 & \frac{1}{\mathrm{c}}\end{array}\right]=\left[\mathrma−1000b−1000\mathrmc−1\begin{array}{ccc}\mathrma^{-1} & 0 & 0 \\ 0 & \mathrm{ b}^{-1} & 0 \\ 0 & 0 & \mathrmc^{-1}\end{array}\right]
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