# Homework Help: Linear Algebra - invertible matrix; determinants

1. Nov 10, 2008

### lubricarret

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

Prove that
[1 a b
-a 1 c
-b -c 1]
is invertible for any real numbers a,b,c

2. Relevant equations

A is invertible if and only if det[A] does not equal 0.

3. The attempt at a solution

I'm not sure if I'm going about this in the correct way;
Would I prove this by solving for the determinant? I did this by cofactor expansion, and came up with:
(1+c^2) - a(-a+bc) + b(ac+b)
= a^2 + b^2 + c^2 + 1

Could I just say then, that the determinant could never be zero, since
a^2 + b^2 + c^2 + 1
will always be nonzero for any real numbers a,b,c?

If someone could just let me know if I did this correctly, or if there is more I need to show.

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

2. Relevant equations

3. The attempt at a solution

2. Nov 10, 2008

### CompuChip

Looks like you got it completely right.
In fact you have shown that always det(A) > 1.

3. Nov 10, 2008

Thanks!