I need to prove that for a square matrix (nXn) A: if rkA= n-1 then rk adjA= 1.

Thanks for any help...

Printable View

- Dec 29th 2009, 02:52 AMrebeccaAdjoint
I need to prove that for a square matrix (nXn) A: if rkA= n-1 then rk adjA= 1.

Thanks for any help... - Dec 29th 2009, 03:33 AMHallsofIvy
The adjoint of a matrix, A, can be defined as the matrix such that $\displaystyle (Au)\cdot v= u \cdot(adj A v)$.

If A has rank n- 1, then it had "nullity" n- (n-1)= 1. That is, the set of all vectors u, such that Au= 0, is a one dimensional subspace. In particular, for any such u, $\displaystyle (Au)\cdot v= 0= u\cdot(adj A v)$.