Consistency of A System of Linear Equations

[Bashyboy]

Hello everyone,

I was just solving a problem in which I had to determine the system of linear equations were consistent. Evidently, if a system of linear equations is capable of being put into triangular form, with no contradictions present, then it must consistent. My question is, why is that so, why does being in triangular form imply consistency?

 

[Simon Bridge]

You should be able to work it out - what does it mean for the system of equations to be "consistent"?

If the system is represented by a triangular matrix, what is the form of the corresponding equations?
Would these be consistent?

Can you see that a diagonal matrix means consistency?
Are there any triangular matrixes that cannot be diagonalized?