Trying to Prove three 2x2 Matrices are Linearly Independent

Hey,

Ive got a question asking;

The set of M_{2,2 }of all 2x2 matrices, with real entries, is a vector space. For what values of A (real) is the set

Z= {(
,

,

}

a linearly independent subset of M_{2,2}?

I have been trying to prove using a_{1}V_{1}+a_{2}v_{2}+a_{3}v_{3}=0 but I cant figure how to prove it or put it into a singular matrix form for row reduction... any help would be greatly appreciated.

Re: Trying to Prove three 2x2 Matrices are Linearly Independent

So you have

so we must have , , and **only** if .

Okay, have you tried **solving** for , , and ? If you subtract twice the first equation from the second, you eliminate - . If you subtract the third equation from the first you again eliminate - . Now eliminate by subtracting twice the first of those two equations from the other: .

If it follows that and . What is the only value of A such that does not have to be 0?