State if determinant of matrix is dependent on unknown?line 1: sqrt2-x _ 1 line2: 1_ sqrt2+x
A two by two determinant of the form |(a, b), (c,d)| is said to be dependent if either one raw can be obtained from the other multiplying with a constant, either one column can be obtained from the other in the same way. In this case the value of the determinant is zero.
(a,,b) = Constant*(c,d)
In general for a determinant of type |(a,b),(c,d)| its value is:
|(a,b), (c,d)| =a*d -b*d
which gives for the determinant in the problem the value:
`|sqrt(2)-x` `1|` = `(sqrt(2)^2 -x^2) -1 =1-x^2`
`|1` `sqrt(2) +x|` ` `
The determinant is zero, (hence dependent) for
`x=1` or `x=-1`
For the rest of `x in (-oo, -1)uu(-1,1)uu(1,+oo)` the determinant is just unknown (in what this means that the value of `x` need to be specified to obtain a certain nonzero value for the determinant).