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If $A=\begin{bmatrix}0&1&-1\\2&1&3\\3&2&1\end{bmatrix}$, then $(A (adj\, A) A^{-1}) A=$ |
$2\begin{bmatrix}3&0&0\\0&3&0\\0&0&3\end{bmatrix}$ $\begin{bmatrix}-6&0&0\\0&-6&0\\0&0&-6\end{bmatrix}$ $\begin{bmatrix}0&1/6&-1/6\\2/6&1/6&3/6\\3/6&2/6&1/6\end{bmatrix}$ none of these |
$2\begin{bmatrix}3&0&0\\0&3&0\\0&0&3\end{bmatrix}$ |
We have, $|A|=\begin{vmatrix}0&1&-1\\2&1&3\\3&2&1\end{vmatrix}=0+7-1=6$ $∴(A (adj\, A) A^{-1}) A=(A (adj\, A))(A^{-1}A)$ $⇒(A (adj\, A) A^{-1}) A=(|A|I)I$ $⇒(A (adj\, A) A^{-1}) A=|A|I=\begin{bmatrix}6&0&0\\0&6&0\\0&0&6\end{bmatrix}=2\begin{bmatrix}3&0&0\\0&3&0\\0&0&3\end{bmatrix}$ |
