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If $I$ is the unit matrix of order $n$. Where$K\neq 0$ is a constsnt , then adj (KI)=

\[\begin{array}{1 1}(1) K^{n}(adj I)&(2)K(adj I)\\(3)K^{2}(adj(I))&(4)K^{n-1}(adj I)\end{array}\]

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I is a unit matrix of order n.
$KI= \begin{bmatrix} K & 0 & 0 &.... & 0 \\0 & K & 0 & .... & 0 \\ | &.... & .... & .... & | \\ 0 & 0 & 0 & .... & K \end{bmatrix}$
Co factor matrix of $KI$
$\qquad= \begin{bmatrix} K^{n-1} & 0 & 0 &.... & 0 \\0 & K^{n-1} & .... & .... & 0 \\ | &.... & .... & .... & .... \\ 0 & 0 & 0 & .... & K^{n-1} \end{bmatrix}$
$\qquad=K^{n-1} \begin{bmatrix} 1& 0 & 0 &.... & 0 \\0 & 1 & .... & .... & 0 \\ | &.... & .... & .... & .... \\ 0 & 0 & 0 & .... & 1 \end{bmatrix}$
$K^{n-1}(adj I)$
Hence 4 is the correct answer.
answered May 5, 2014 by meena.p
 

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