If B is the inverse of a third order matrix A | vedclass

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(C) Given that $A$ is a matrix of order $n = 3$ and $B = A^{-1}$.We know that $\det B = \det(A^{-1}) = \frac{1}{\det A} = k$,so $\det A = \frac{1}{k}$

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$k B$

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(C) Given that $A$ is a matrix of order $n = 3$ and $B = A^{-1}$.We know that $\det B = \det(A^{-1}) = \frac{1}{\det A} = k$,so $\det A = \frac{1}{k}$.The property for the adjoint of an adjoint matrix is $\operatorname{adj}(\operatorname{adj} A) = (\det A)^{n-2} A$.Since $n = 3$,we have $\operatorname{adj}(\operatorname{adj} A) = (\det A)^{3-2} A = (\det A) A$.Substituting $\det A = \frac{1}{k}$,we get $\operatorname{adj}(\operatorname{adj} A) = \frac{1}{k} A$.Now,we need to find the inverse: $(\operatorname{adj}(\operatorname{adj} A))^{-1} = (\frac{1}{k} A)^{-1}$.Using the property $(c A)^{-1} = \frac{1}{c} A^{-1}$,we get $(\frac{1}{k} A)^{-1} = k A^{-1}$.Since $A^{-1} = B$,the expression becomes $k B$.

If $B$ is the inverse of a third order matrix $A$ and $\det B = k$,then $(\operatorname{adj}(\operatorname{adj} A))^{-1} =$

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