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Clearly |A|=0 ,so A*adj A=0 now since A is not null therefore Adj A can be anything , is may or may not be null so how can we say directly that adj A is not equal to 0
 
 
 
 

1 Answer

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If the Rank of matrix  A is n-1 then there is atleast one minor present of order n-1 of the matrix A is not equal to zero.Therefore the matrix Adj A will be a non zero matrix  and thus the Rank of the matrix Adj A will be greater than zero.

Now 

Rank of matrix A is n-1 Therefore | A | =0

Therefore A ( Adj A) will be a Zero matrix.

and we know

If the Rank of matrix  A is n-1 then there is atleast one minor present of order n-1 of the matrix A is not equal to zero.

Hence we can say directly Adj A will not be Zero.

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