The expression provided is:

$\text{det} = x \cdot \left( x + y - \frac{y^2}{x} \right) \cdot (\text{third diagonal element})$

where the third diagonal element is given as $y + x$.

Substituting the third diagonal element:

We replace "third diagonal element" with $y + x$, which gives:

$\text{det} = x \cdot \left( x + y - \frac{y^2}{x} \right) \cdot (y + x)$

Simplifying the expression inside the parentheses:

The term $x + y - \frac{y^2}{x}$ can be combined:

$x + y - \frac{y^2}{x} = \frac{x^2 + yx - y^2}{x}$

Now the determinant expression becomes:

$\text{det} = x \cdot \frac{x^2 + yx - y^2}{x} \cdot (y + x)$

Canceling out $x$ in the numerator and denominator:

$\text{det} = (x^2 + yx - y^2) \cdot (y + x)$

Expanding the product:

$\text{det} = (x^2 + yx - y^2)(y + x)$

Using the distributive property:

$\text{det} = x^2 \cdot y + x^2 \cdot x + yx \cdot y + yx \cdot x - y^2 \cdot y - y^2 \cdot x$

This expands to:

$\text{det} = x^2y + x^3 + y^2x + yx^2 - y^3 - y^2x$

Simplifying like terms:

Notice that $y^2x$ cancels out:

$\text{det} = x^2y + x^3 + yx^2 - y^3$

The final expression for the determinant is:

$\boxed{\text{det} = x^3 + 2x^2y - y^3}$

Let me know if you would like further clarification on any of the steps or need additional details.

Relative Questions:

1. How do you apply the distributive property when expanding polynomials?
2. Can you explain how to identify and cancel like terms in algebraic expressions?
3. How can this determinant be used in a larger matrix context?
4. What is the significance of diagonal elements in matrices?
5. How would this expression change if the third diagonal element were different?

Tip:

When simplifying complex algebraic expressions, always look for common terms to cancel out early to avoid unnecessary expansions.