# Matrices, Equivalent Conditions

• Sep 27th 2009, 01:11 PM
Alterah
Matrices, Equivalent Conditions
My problem is this:
State whether the following is true or false, provide a reason.
1. Ax = O has only the trivial solution if and only if Ax = b has a unique solution for every n x 1 column matrix b.

I am leaning towards false. The reason for this lies with a list of equivalent conditions in my book:
If A in an n x n matrix, then the following statements are equivalent:
1. A is invertible
2. Ax = b has a unique solution for every n x 1 column matrix b
3. Ax = O has only the trivial solution
4. A is row equivalent to \$\displaystyle I_n\$
5. A can be written as a product of elementary matrices.

The reason I am leaning towards false is because the problem makes it seem that part one of it (Ax = O ...) can only be true if part two (Ax = b...) is. Perhaps I am misinterpreting however. Thanks for input.
• Sep 27th 2009, 10:05 PM
NonCommAlg
Quote:

Originally Posted by Alterah

My problem is this:
State whether the following is true or false, provide a reason.
1. Ax = O has only the trivial solution if and only if Ax = b has a unique solution for every n x 1 column matrix b.

it's true because \$\displaystyle A\bold{x} = \bold{o}\$ has only the trivial solution \$\displaystyle \bold{x}=\bold{0}\$ if and only if \$\displaystyle A\$ is invertible and so \$\displaystyle \bold{x}=A^{-1}\bold{b}\$ is the unique solution of \$\displaystyle A\bold{x}=\bold{b}.\$