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

In summary, the conversation discusses finding the determinant of a given matrix A and determining the values of x, y, and z that make the matrix not invertible. The determinant is found to be equal to -z^2-x^2, and it is stated that the matrix is invertible when the determinant is zero. The question arises of how to list all possible values that make the determinant zero, and it is suggested to prove that there are only finitely many possibilities. Additionally, it is mentioned that y can be any real number as long as the equation is satisfied.
  • #1
Taryn
63
0
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?
 
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  • #2
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)
 
  • #3
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!
 
  • #4
Taryn said:
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 }.
 
  • #5
ahhh okay that helps... thanks!
 

1. How do you solve for x, y, and z in a matrix determinant?

The most common method used to solve for x, y, and z in a matrix determinant is by using Gaussian elimination. This involves performing row operations to reduce the matrix to its reduced row echelon form, from which the values for x, y, and z can be easily determined.

2. What does it mean for a matrix to be invertible?

A matrix is considered invertible if its determinant is non-zero. This means that the matrix has a unique solution and can be inverted to obtain its inverse matrix, which can be used to solve for variables in a system of equations.

3. Can you use the determinant to determine if a matrix is invertible?

Yes, the determinant is a useful tool for determining if a matrix is invertible. If the determinant is non-zero, then the matrix is invertible. If the determinant is zero, then the matrix is not invertible.

4. What is the formula for calculating the determinant of a 3x3 matrix?

The formula for calculating the determinant of a 3x3 matrix is:
|A| = a11(a22a33 - a23a32) - a12(a21a33 - a23a31) + a13(a21a32 - a22a31)

5. Is the determinant the only way to solve for variables in a system of equations?

No, the determinant is not the only way to solve for variables in a system of equations. Other methods include using matrices to represent the system and solving using Gaussian elimination, or using Cramer's rule to solve for individual variables.

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