Can someone show me the proof in Linear Algebra? A* [adj(A)] = det(A)* A?

Question

A is a nxn matrix.

Answer 1

Actually, it's A*[adj(A)] = det(A)*I

Here's the main idea, at least:

Let B be the matrix on the right side, det(A)*I. Then the (i,j) entry of B is the sum

sum k=1 to n Aik*(-1)^(k+j) det(Cjk)

where Cjk is the matrix with row j and column k deleted.

If i=j, then this is the determinant of A.

If i is not equal to j, then this is the determinant of a matrix obtained by replacing row j of A with row i. The result is a matrix with two identical rows, which is singular, and therefore has determinant 0.

Answer 2

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