# Inverse of matrix with unknown values

1. Sep 30, 2012

### ryan1180

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

Let B=$$\begin{pmatrix} λ-3 & 12 & -1 \\ 0 & λ+2 & λ \\ 0 & 0 & 5 \end{pmatrix}$$

For what values of λ does matrix B have an inverse?

2. Relevant equations

3. The attempt at a solution

I first tried to calculate det(B)=(λ-3)(5λ+10) +0 -0

to get x=-2 or 3.

I'm unsure if this calculation helps me, and what to do beyond this point.

2. Sep 30, 2012

### Dick

Well, if det(B)=0 then it doesn't have an inverse. If det(B) is nonzero, then it does. So?

3. Sep 30, 2012

### jbunniii

Yes, this calculation helps you, because the matrix is invertible if and only if det(B) is nonzero. Your calculation tells you exactly what values of $\lambda$ make det(B) zero.

4. Sep 30, 2012

### ryan1180

Okay, so since the matrix is singular when λ=-2 or 3, my answer then be that the matrix has an inverse at λ≠-2,3, correct?

5. Sep 30, 2012

### Dick

Correct.

6. Sep 30, 2012

### ryan1180

Thanks! It looks like I just failed to make a simple connection between concepts.

There is also a part B that asks: How many solutions will the system Bx = 0 have if λ = 2?
I've tried to find an answer for this part, but haven't had much success. Looking ahead in my book seems to indicate that finding the "nullspace" will help to solve that. Since we haven't done that yet, is there any other way that I would be able to solve that part?

7. Sep 30, 2012

### Dick

You've decided B is invertible if λ = 2, right? Multiply both sides of Bx=0 by B^(-1).

8. Sep 30, 2012

### Ray Vickson

Question: if you were unsure whether the determinant calculation could help you, why did you do it?

RGV