https://file.aiquickdraw.com/mathbot/file/172820800556745frake7.jpg

The matrix from your image is:

\[
\begin{pmatrix}
3x + 2x & 3x + 2x & 3x + 3y & 3x + 2y \\
2y + z & 3y + z & z - 1 & z + 1 \\
\end{pmatrix}
\]

To check its consistency, let’s simplify the elements.

### First row:
1. \( 3x + 2x = 5x \)
2. \( 3x + 2x = 5x \)
3. \( 3x + 3y \) (already simplified)
4. \( 3x + 2y \) (already simplified)

Thus, the first row simplifies to:
\[
(5x, 5x, 3x + 3y, 3x + 2y)
\]

### Second row:
1. \( 2y + z \) (already simplified)
2. \( 3y + z \) (already simplified)
3. \( z - 1 \) (already simplified)
4. \( z + 1 \) (already simplified)

Thus, the second row simplifies to:
\[
(2y + z, 3y + z, z - 1, z + 1)
\]

### Simplified matrix:
\[
\begin{pmatrix}
5x & 5x & 3x + 3y & 3x + 2y \\
2y + z & 3y + z & z - 1 & z + 1 \\
\end{pmatrix}
\]

#### Consistency:
To test the consistency of this matrix, it needs to be checked if any row results in a contradiction or if the matrix has a unique solution. This can involve methods like Gaussian elimination or checking for linear dependence between rows. Here, the rows are different, so there's no obvious contradiction.

If you have specific values for \( x \), \( y \), and \( z \), you can plug them in and verify if the system holds. Alternatively, we can analyze further for linear dependence or perform operations to reduce the matrix to an echelon form.

Would you like me to continue with Gaussian elimination or a specific operation?

---

**Related questions:**
1. How do you perform Gaussian elimination on a matrix?
2. What does it mean for a system of equations to be consistent?
3. How do you test for linear dependence between rows of a matrix?
4. What are the implications of having a row of zeroes in a matrix?
5. Can you provide examples of inconsistent systems?

**Tip:** In matrix analysis, checking for consistency involves verifying if the system of equations can be solved or leads to contradictions.

Learn how to test the consistency of a matrix by simplifying rows and using Gaussian elimination. This detailed solution covers key concepts like linear dependence, matrix consistency, and Gaussian elimination. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation