# Homework Help: Linear algebra - adjoint matrix = 0

1. Jan 30, 2010

### Kate2010

1. The problem statement, all variables and given/known data

Ok so the question is:

Let A be an nxn matrix where n $$\geq$$ 2. Show A* = 0 (A* is the adjoint matrix, 0 is the zero matrix) if and only if rank(A)$$\leq$$ n-2.

2. Relevant equations

A*:= ((-1)i+jdet(Aij))ij where Aij is the (i,j) minor of A

3. The attempt at a solution

I'm really struggling with answering this.

If rank(A) $$\leq$$ n-2 then A contains at least 2 dependent rows. Does this automatically make A* = 0?

If A*=0 then (-1)i+jdet(Aij) = 0 for all i and j. So det(Aij) = 0 for all i and j. Having a zero determinant implies dependence?