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a) In this sense, develop the calculations and determine the quantities of defective parts x, y and z of this machine in one hour.

b) What is the characteristic of a diagonal matrix? Give an example.

Let's tackle both parts of your problem based on the matrix provided:

### a) Determine the quantities \( x \), \( y \), and \( z \) (defective parts per hour):

The matrix provided is:

\[
A = \begin{bmatrix} 
x & 0 & 0 \\
z - 1 & y & x^2 - 6x + 9 \\
0 & y - 3 & z 
\end{bmatrix}
\]

Given that the problem asks to determine the values of \( x \), \( y \), and \( z \), we need more context or additional conditions (such as values or constraints on the matrix). This question likely involves solving the matrix based on some missing real-world interpretation, perhaps related to defective parts in production.

However, from the structure of the matrix, we can analyze it algebraically:

1. **Entry (3, 3):** The element \( z \) appears directly. If we had more information (such as a determinant or other constraint), we could solve for \( z \).
   
2. **Entry (1, 1):** The element \( x \) is isolated. Thus, once we know \( x \), it directly applies to the defective parts rate.
   
3. **Row 2, Column 3:** The element \( x^2 - 6x + 9 \) is a quadratic expression, which simplifies to:
   \[
   x^2 - 6x + 9 = (x - 3)^2
   \]
   This suggests that the expression will be zero when \( x = 3 \).

To fully determine \( x \), \( y \), and \( z \), we would need additional conditions or numerical constraints. If there are missing details in the problem setup (e.g., the matrix being diagonal, singular, or having a determinant equal to a certain value), please provide that, and we can proceed further.

### b) Characteristic of a Diagonal Matrix:

A **diagonal matrix** is a matrix in which all the non-diagonal elements are zero. This means that every entry \( a_{ij} \) is zero when \( i \neq j \). Only the elements along the main diagonal (from the top left to the bottom right) can be non-zero.

#### Example of a diagonal matrix:
\[
D = \begin{bmatrix}
2 & 0 & 0 \\
0 & 5 & 0 \\
0 & 0 & 3
\end{bmatrix}
\]

In this matrix:
- The elements \( 2 \), \( 5 \), and \( 3 \) lie on the diagonal.
- All other elements are zero.

This property simplifies many matrix operations, such as matrix multiplication and finding determinants (which for diagonal matrices is the product of the diagonal elements).

---

Do you want further details or have any questions on this?

Here are five related questions you might want to explore:
1. How do you compute the determinant of a matrix like the one provided?
2. What are the conditions for a matrix to be invertible?
3. How does matrix multiplication behave with diagonal matrices?
4. What is the rank of a matrix, and how does it relate to the values of \( x \), \( y \), and \( z \) in this case?
5. How can eigenvalues be computed for a diagonal matrix?

**Tip:** When working with quadratic expressions like \( x^2 - 6x + 9 \), always check if they can be factored into perfect squares, as this simplifies solving.

a) In this sense, develop the calculations and determine the quantities of defective parts x, y, and z of this machine in one hour.
b) What is the characteristic of a diagonal matrix? Give an example.

In this problem, explore how to solve a matrix for variables x, y, and z, while analyzing quadratic expressions. Additionally, understand the characteristics of a diagonal matrix and review an example. Suitable for students in Grades 10-12.

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