Ok so the question is:
Let A be an nxn matrix where n 2. Show A* = 0 (A* is the adjoint matrix) if and only if rank(A) n-2.
I'm really struggling with answering this. These are my thoughts:
If rank(A) 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?
Linear algebra - adjoint matrix = 0 Watch
- Thread Starter
- 30-01-2010 20:30
- 31-01-2010 23:26
I think you mean the adjugate matrix. I don't know the proof of the result, but I can imagine that it is true - almost every minor will obviously have 2 dependent rows or columns; the ones that don't obviously have dependent rows or columns lie on the diagonal.
Yes, if the determinant of a matrix is zero then there is a dependent row or column.