Q70 / 180 · KCET 2012 · Mathematics

Inverse of a diagonal non-singular matrix is

  • A
    scalar matrix
  • B
    skew symmetric matrix
  • C
    zero matrix
  • Correct answer
    D
    diagonal matrix

Solution

Let diagonal matrix $A=\left[\begin{array}{lll}a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a\end{array}\right]$ Now, $\quad|\mathrm{A}|=\mathrm{a}\left(\mathrm{a}^{2}\right)=\mathrm{a}^{3}$ $\therefore$ $\operatorname{adj} A=\left[\begin{array}{ccc}a^{2} & 0 & 0 \\ 0 & a^{2} & 0 \\ 0 & 0 & a^{2}\end{array}\right]$ $\begin{aligned} \therefore \mathrm{A}^{-1}=\frac{\operatorname{adj}(\mathrm{A})}{|\mathrm{A}|} &=\frac{1}{\mathrm{a}^{3}}\left[\begin{array}{ccc}\mathrm{a}^{2} & 0 & 0 \\ 0 & \mathrm{a}^{2} & 0 \\ 0 & 0 & \mathrm{a}^{2}\end{array}\right] \\ &=\left[\begin{array}{ccc}1 / \mathrm{a} & 0 & 0 \\ 0 & 1 / \mathrm{a} & 0 \\ 0 & 0 & 1 / \mathrm{a}\end{array}\right] \end{aligned}$ which is an diagonal matrix.

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

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