Let $A = \begin{bmatrix} 1 & 1 & 2 \ - 2 & 0 & 1 \ 1 & 3 & 5 \end{bmatrix}

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Let $A = \begin{bmatrix} 1 & 1 & 2 \ - 2 & 0 & 1 \ 1 & 3 & 5 \end{bmatrix}$. Then the sum of all elements of the matrix $adj\left( adj\left( 2(adj	ext{ }A)^{- 1} ight) ight)$ is equal to :

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Sol. [Key Concept: Use $(adjA)^{- 1} = A/|A|$ and $adj(adjB) = {|B|}^{n - 2}B$ for a $3 	imes 3$ matrix.]$$|A| = - 4 $$As $(adjA)^{- 1} = \frac{A}{|A|} = \frac{- A}{4}$$${B = 2(adjA)^{- 1} = \frac{- A}{2} }{|B| = \left| \frac{- A}{2} ight| }{= \frac{- 1}{8}|	ext{ }A| = \frac{1}{2} }{adj(adjB) = |B|^{n - 2}	ext{ }B = |B|B }{= \frac{B}{2} = \frac{- A}{4} }$$Sum of element $= \frac{- 12}{4} = - 3$

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