if A is a matrix of order 3*3 and Iadj A I equal to 81, what will be the sum of the product of elements of any row of A with their corresponding cofactors
 

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Please find below the solution to the asked query:

We know that:AdjA=An-1, where 'n' is the order of 3×3 matrix.Here n=3Given that AdjA=81A3-1=81A2=81A=9Now we know that:Sum of the prodcuts of the elements of any row or column with their corresponding cofactors in a determinant is equal to the value of the determinant.Hence required answer is 9.
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ladjAl=lAl raised to n-1 where n is the order
So,
81=lAl³-¹
81=|A|²
9=|A|
Also we know that sum of product of elements of any row of a matrix with its corresponding cofactors is the determinant of that matrix.
So the answer is 9
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