Solve Matrix Determinant: Find x,y,z for Invertibility

[Taryn]

I have just tried to solve this problem and just wondering if I am right!

1) Compute the determinant of the matrix A
-1 -1 1
x^2 y^2 z^2
0 -1 0
and find all real numbers x,y, and z such that A is not invertible.

Okay so I found that the det=-z^2-x^2
So when the matrix is invertible the determinant is zero!

-z^2-x^2=0

Can I say that matrix is invertible when z^2=-x^2?
So my question from here is would I just list numbers that would make the det zero? And how would I find y?

 

[Hurkyl]

So when the matrix is invertible the determinant is zero!

Assuming you meant "not" invertible, your work looks right. There's a little bit more to do, though.

So my question from here is would I just list numbers that would make the det zero?

If you mean just write down examples, then no. You need to write down the set of all possibilities! But, if you can prove that there are only finitely many possibilities, then writing them all down is good enough.

And how would I find y?

You choose y so that the equation is satisfied. (hint: it's easy. You're probably overthinking it)

 

[Taryn]

Sorry not following with the last part!
You choose y so that the equation is satisfied??
U mean I substitute y in for x or somethin!

 

[radou]

Sorry not following with the last part!
You choose y so that the equation is satisfied??
U mean I substitute y in for x or somethin!

Since the determinant doesn't depend on y, y can be any real number.

Hurkyl probably meant something like: S = {(x, y, z) E R^3 : z^2 = - x^2 & y E R }.

 

[Taryn]

ahhh okay that helps... thanks!