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If $A$ is a scalar matrix with scalar $k \neq 0$ of order $3$ than $A^{-1}$ is

\[\begin{array} {1 1}(1)\frac{1}{k^{2}}I&(2)\frac{1}{k^{3}}I\\(3)\frac{1}{k}I&(4)kI\end{array}\]

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$A=\begin{bmatrix} K & 0 & 0 \\ 0 & K & 0 \\ 0 & 0 & K \end{bmatrix}$
$A^{-1}= \large\frac{1}{|A|} $$ adj A$
Co factor matrix of $A=\begin{bmatrix} K^2 & 0 & 0 \\ 0 & K^2 & 0 \\ 0 & 0 & K^2 \end{bmatrix}$
$|A|=K^3$
$A^{-1}=\large\frac{1}{K^3} $$ K^2 \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$
$\qquad =\large\frac{1}{K} $$I$
Hence 3 is the correct answer.
answered May 2, 2014 by meena.p
 

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