An linear algebra question (not an newb quesiton)?

Question

for the 3x3 matrix ( ";" represents the end of a row)
[c,1,0; 1,c,1; 0,1,c], find all values of c, if any, that would make the matrix invertible. okay my own guess here c is anything but 0 (since 0 will make the first and second row identical, thus creating a all 0 row when u row reduce which means it's not invertible because u cannot obtain the identity matrix by row reducing) I suppose my real quesiton is the way to approach these types of problem in general. My thought is that any number that will create a 0 row when row reduce will obviously be excluded, but am I right in assuming that every other number will work therefore? or is there some other cases i need to look into?

Answer 1

Remember that a matrix is invertible if and only if its determinant is nonzero. So we need to find out what values of c will make the determinant of this matrix 0 and exclude them, and then any other value will work. So finding the determinant:

[c, 1, 0]
[1, c, 1]
[0, 1, c]

determinant = c³ + 0 + 0 - 0 - c - c = c³ - 2c

Setting this equal to zero:

c³ - 2c = 0
c = 0 ∨ c² - 2 = 0
c = 0 ∨ c = √2 ∨ c = -√2.

So any value of c other than 0 or ±√2 will work.

Answer 2

I just tried quite a few values of c, positive and negative in matlab it is seems like they all work. You are right 0 will not work.

sqrt(2) does not work, I think because the determinate is zero.