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I'm not sure this should go in homework of here as this was a test question, but the question its self isn't a test question.

I got this question marked wrong, for the record.

**Question**

__For each statement below, determine if the statement is true or false. If true, provide a proof, if false, provide a counterexample.__

A.) If A is a nonzero matrix, and if AB = 0 (the zero matrix), then B = 0

Suppose [tex]A[/tex] has a inverse and[tex] B ≠ 0 [/tex]

Then,

[tex] A^{-1}AB = 0{A^-1}[/tex]

[tex]I_nB = 0[/tex]

Since Identity * B ≠ 0 unless B is the zero matrix, B must be zero.

However, the professor simply wrote false and then gave an example of how it was possible for the matrix to not be the zero matrix.

However, I clearly showed B can be zero.

So, the real question I'm asking is if this question was to be seen in any sense what is the context of the equal sign; in terms of any mathematics to make a statement false of true.

For example would this statement also be false..in terms of the question below.

__For each statement below, determine if the statement is true or false. If true, provide a proof, if false, provide a counterexample.__

Let f(x) = 4 and f(x) = x^2, then x = -2.

I would have to say false?

Because,

Let f(x) = 4 and f(x) = x^2, then x = -2 and x = 2.

Comments please, and rip it apart :D