# Thread: Linear independence & inversion question

1. ## Linear independence & inversion question

"Show that if ad - bc = 0, then the equation Ax = 0 has more than one solution. Why does this imply that A is not invertible?"

The problem then gives a hint about considering when a=b=0. That's pretty straightforward--the matrix becomes linearly dependent. But what to do after that? What specifically does this question want?

2. ## Re: Linear independence & inversion question

linearly DEpendent.

we always know that (0,0) is a solution. so if we have any OTHER solution, A cannot be invertible.

well, if a = b = 0, then if c = 0, we have the non-zero solution (1,0), and if d = 0, we have the non-zero solution (0,1).

otherwise, we have the non-zero solution (1,-c/d).

so suppose a is non-zero. then we have the non-zero solution (-b/a,1).

why does this show A is non-invertible?

if we have Ax = 0, with x0 then what do we choose for the value A-1(0)?

3. ## Re: Linear independence & inversion question

In general if Ax= b (not necessarily equal to 0) and A has an inverse then it follows that $x= A^{-1}b$ so the equation has only that one solution. Any time there is more than one solution (or no solution) A must have no inverse.