Proving Invertible Matrices: A and B are n × n Matrices

[charlies1902]

Let A and B be n × n matrices.
a. Show that if A is invertible and AB = 0, then
B = 0.



If A is invertible, it can be reduced to the I matrix.
Thus IB=0 (this is the part where I'm hesitant, can I say that IB=0?)
Thus B=0 since I≠0

 

[jbunniii]

Let A and B be n × n matrices.
a. Show that if A is invertible and AB = 0, then
B = 0.



If A is invertible, it can be reduced to the I matrix.
Thus IB=0 (this is the part where I'm hesitant, can I say that IB=0?)
Thus B=0 since I≠0

How did you reduce A to the I matrix? What operation(s) did you perform?

 

[charlies1902]

How did you reduce A to the I matrix? What operation(s) did you perform?

Multiplying A^-1 to both sides. Is this necessary to state in the proof?

When I got the BI=0, I think I got it for the wrong reasons. I'm not very good at proofs. It took me awhile to think of multiplying A^-1 to both sides that that's how it justified BI=0. Typically when you go about doing proofs, how do you think in the "right" direction? Proofs seem to be so open ended compared to computation problems, that if you're not thinking in the "right" direction, it might take you a very long time to get it. Do you have any tips?

 

[HallsofIvy]

Multiplying A^-1 to both sides. Is this necessary to state in the proof?

Yes, it is necessary! That is, in fact, exactly what you are doing.

When I got the BI=0, I think I got it for the wrong reasons. I'm not very good at proofs. It took me awhile to think of multiplying A^-1 to both sides that that's how it justified BI=0.

No, you do NOT get "BI= 0". You had AB= 0 and you multiply each side by A on the left (remember that matrix multiplication is NOT commutative) so you get "A^{-1}(AB)= A^-1A)B (matrix multiplication IS associative)= IB= A^{1}0.

Typically when you go about doing proofs, how do you think in the "right" direction? Proofs seem to be so open ended compared to computation problems, that if you're not thinking in the "right" direction, it might take you a very long time to get it. Do you have any tips?

Yes, proofs require thinking rather than just following what you were told to do. To do that you going to have to know the precise statement of definitions and other theorems and use the to "build a bridge" from the hypothesis to the conclusion.