Calculate the determinant of a 3x3 matrix with variables x, y, and z.

Published on February 23, 2025

Linear-Algebra

The determinant is evaluated by splitting it into two parts, factoring out terms, and using the Vandermonde determinant formula. The final result is expressed as a product of factors involving x, y, and z, and a term (1-xyz).

Question

Answer

We are asked to evaluate the determinant:

We can split the determinant along the third row:

Let

and

. Then

For

, we can factor out

from the first column,

from the second column, and

from the third column:

This is a Vandermonde determinant, and its value is

.
So,

For

, we can factor out

from the third row:

Swap the first and third rows:

Swap the second and third rows:

Thus,

Therefore,

.
Since

, we have

.
So,

Final Answer: The final answer is

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