Question

probe that the product of a singular matrix with its adjoint is the null matrix.​

Answer 1

Answer:

Is the adjoint of a singular matrix always the zero matrix?

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Alex Eustis

Answered 1 year ago

Is the adjoint of a singular matrix always the zero matrix?

The preferred term is Adjugate matrix - Wikipedia.

The adjugate of any matrix is defined as the transpose of the cofactor matrix. In the 2x2 case, if

A=[acbd]

then

adj A=[d−c−ba]

and clearly this is the zero matrix only if A itself is.

More generally, for an n×n matrix A , the rank of A is the largest number r such that at least one r×r submatrix has a nonzero determinant. A proof of this fact may be found here: Proof that determinant rank equals row/column rank

From this it follows that adj A=0 if and only if rank A≤n−2 . Just observe that the entries of adj A are all of the (n−1)×(n−1) determinants of A with a possible minus sign.