Let A be a square matrix all of whose entries | vedclass

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(C) The inverse of a square matrix $A$ is given by the formula: $A^{-1} = \frac{1}{\det(A)} 	ext{adj}(A)$. Since $A$ is a square matrix with integer

Vedclass · Accepted Answer

If $\det(A) = \pm 1$ then $A^{-1}$ exists and its entries are integers.

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(C) The inverse of a square matrix $A$ is given by the formula: $A^{-1} = \frac{1}{\det(A)} 	ext{adj}(A)$. Since $A$ is a square matrix with integer entries,the adjoint matrix $	ext{adj}(A)$ also consists entirely of integers (as it is the transpose of the cofactor matrix). If $\det(A) = \pm 1$,then $A^{-1} = \frac{1}{\pm 1} 	ext{adj}(A) = \pm 	ext{adj}(A)$. Since $	ext{adj}(A)$ contains only integers,$\pm 	ext{adj}(A)$ also contains only integers. Therefore,if $\det(A) = \pm 1$,$A^{-1}$ exists and all its entries are integers.

Let $A$ be a square matrix all of whose entries are integers. Then which one of the following is true $?$

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