I have this proof and I am wondering if it is complete. The question trying to b
ID: 1945514 • Letter: I
Question
I have this proof and I am wondering if it is complete.The question trying to be proved is: Prove the product of an invertible matrix and a singular matrix is singular.
If matrix A is invertible and matrix B is singular, then let X be a nonzero vector such that BX = 0. Then ABx = A0 = 0 So AB has a nontrivial null vector, and therefore is singular.
I'm trying to figure out if this is completed. If anyone has any ideas of how I can make this a finished proof I would appreciate it. or if this is completed how would I explain it.
Explanation / Answer
It is almost complete... you showed that AB is singular, but you haven't showed that BA is singular (I think they are requiring both) to finish, since A is invertible, there exists matrix such that AC=1. Let Y=CX (X defined as before). Y can't be 0 (since AY=ACX = X and X is not 0). Then BAY=BA(CX) = BACX = B1(X) = BX = 0 Thus BA is singular.