State if determinant of matrix is dependent on unknown?line 1: sqrt2-x _  1 line2: 1_ sqrt2+x

Expert Answers
valentin68 eNotes educator| Certified Educator

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).

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