State if determinant of matrix is dependent on unknown?

line 1: sqrt2-x _ 1

line2: 1_ sqrt2+x

### 1 Answer | Add Yours

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

`1-x^2 =0`

`(1-x)(1+x) =0`

`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).

### Join to answer this question

Join a community of thousands of dedicated teachers and students.

Join eNotes