Determine the values of for which the 3x3 matrix below is

Determine the values of ? , for which the 3x3 matrix below is invertible:

?-1 0

-1 ? -1

0 -1 ?

Note: Please show detailed work.... Also any ? =lambda sign.

Replace lamb by a variable say \'x\' :

Our matrix looks like this :

x -1 0
-1 x -1
0 -1 x

Let us find the determinant :

x(x^2 - 1) + 1(-x + 0) + 0

x^3 - x - x = 0

x^3 - 2x = 0

x(x^2 - 2) = 0

Solving by zero factor property :

x = 0 and x^2 - 2 = 0

x = 0 and x = +/-sqrt2

So, for these values of x(i.e lambda), the matrix is non-invertible

So, in conclusion,

Any value of lambda other than 0 , sqrt2 or -sqrt2 is the condition that the given matrix is inverible