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If $A= \; \begin{bmatrix} 1& 0 &0 \\ 2 & 3 & 4\\ 0&1 & 2 \end{bmatrix}$, then the value of $A.(Adj\:A)$ is?

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• For a matrix $A$ which is non singular,$A^{-1}=\large\frac{1}{|A|}$$(Adj\:A) • A.A^{-1}=I Given: A= \; \begin{bmatrix} 1& 0 &0 \\ 2 & 3 & 4\\ 0&1 & 2 \end{bmatrix} \Rightarrow det (A)=|A| = (1) \times (2 \times 3 - 1\times 4) + (0) \times (2 \times 2- 0\times 4)+(0)\times ( 2 \times 1-0 \times 3) =6-4=2 \Rightarrow\:A is non singular. We know that for a matrix A which is non singular,A^{-1}=\large\frac{1}{|A|}$$(Adj\:A)$
$\Rightarrow\:Adj\:A=|A|.A^{-1}$
$\Rightarrow\:A\:(Adj\:A)=A.A^{-1}.|A|$
$=I.|A|=2I$
answered Jan 30, 2014
edited Jan 30, 2014