Understanding Linear Homogeneous Systems: Finding the Correct Answers

[DanielJackins]

Homework Statement

If a linear homogeneous system Ax=0 has a non - trivial solution and A is an n x n matrix, then (choose ALL correct answers)

A. A has rank less than n
B. Each system Ax=b with the same coefficient matrix A has a solution
C. A is row equivalent to I
D. If Ax=b has one solution it has many
E. A is invertible

If a linear homogeneous system Ax = 0 has a non - trivial solution and A is n x n, then (choose ALL correct answers)

A. A is invertible
B. A has rank less than n
C. A is row equivalent to I
D. If Ax = b has one solution it has many
E. Each system Ax = b with the same coefficient matrix A has a solution

The Attempt at a Solution

So I looked through my notes for these two questions, found some properties and chose the answers the notes led me to believe were true, but I got both incorrect. I only have one attempt left on each question so I want to make sure I'm 100% sure on the answers Ibefore I try. Could anyone give me a nudge in the right direction? ie. explain linear homogeneous systems?

 

[aPhilosopher]

what were the properties that led you to the incorrect answers? What were those answers?

It would help if I knew what I needed to explain.

 

[DanielJackins]

My notes said that a square matrix is also invertible, and is row equivalent to I

 

[aPhilosopher]

Oh, ok. Only a square matrix can be invertible. Not all of them are. Your notes must be mistaken.

 

[DanielJackins]

So even though it's a square it isn't necessarily invertible?

 

[aPhilosopher]

precisely. Read https://www.physicsforums.com/showpost.php?p=2369970&postcount=22" post. I wrote it for somebody that was really being dense so if it seems like it's talking down to you a little, don't take it personally ;)

It gives the simplest examples possible of a matrix equation where the matrix is not invertible. If you have any questions, I'll be happy to answer.

 

[DanielJackins]

Okay thanks for the link. But I'm still kind of lost about the other choices

 

[defunc]

The only true statement in both cases is that A has rank less than n.

 

[aPhilosopher]

There's one more true statement. Which is of course equivalent to the one that you mentioned.

 

[DanielJackins]

Sorry what? So A has rank less than n, and there is one more true statement other than that?