what is proof ofA*adj(A) = |A|*Iwhere A is a square matrix of order n and I is identity matrix of order n. plz reply with proper derivation.

Dear vikash,

As a consequence of Laplace formula for the determinant of an n×n matrix A, we have

where I is the n×n identity matrix Indeed, the (i,i) entry of the product A adj(A) is the scalar product of row i of A with row i of the cofactor matrix C, which is simply the Laplace formula for det(A) expanded by row i. Moreover, for i ≠ j the (i,j) entry of the product is the scalar product of row i of A with row j of C, which is the Laplace formula for the determinant of a matrix whose i and j rows are equal and is therefore zero.

From this formula follows one of the most important results in matrix algebra: A matrix A over a commutative ring R is invertible if and only if det(A) is invertible in R.

For if A is an invertible matrix then

and if det(A) is a unit then (*) above shows that

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