Two fairly simple proof problems. . . why aren't they simpler? :((adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

Let A be an nxn matrix...

If A is not invertible then there exists an nxn matrix B such that AB = 0, B != 0. (not equal to)

2. Relevant equations

None really.

3. The attempt at a solution

Obviously, when A is the zero matrix, AB = 0.

If we call A the coefficient matrix in the system of equations Ax = 0, then x = x_{1}B_{1}+ x_{2}B_{2}+ ... + x_{n}B_{n}, where B = [B_{1}|B_{2}|...|B_{n}].

I can't seem to explain why that works. Is it obvious enough just to say that or is there a step of explanation I have left out?

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1. The problem statement, all variables and given/known data

If A is an m x n matrix, B is an n x m matrix and n < m, then AB is not invertible.

2. Relevant equations

3. The attempt at a solution

Obviously, A and B are not square and are therefore not invertible. Does that fact really matter? The product of invertible matrices is invertible, but is the product of non invertible matrices also non invertible?

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# Homework Help: Linear Algebra - Proofs involving Inverses

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