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It appears that you've shared an image involving matrix operations, specifically working with non-singular matrices \( P \) and \( Q \), and aiming to find their product to achieve a normal form of a given matrix \( A \).

Let me summarize the process visible in your image:

- **Goal:** Find the non-singular matrices \( P \) and \( Q \) such that \( PAQ \) results in the normal form (likely an identity matrix).
- **Matrix \( A \):** The matrix is clearly shown in the image with the values:
  \[
  A = \begin{bmatrix}
  3 & 2 & 4 & 5 \\
  5 & 1 & 4 & -2 \\
  1 & -4 & 11 & -14
  \end{bmatrix}
  \]
- **Elementary Row and Column Operations:** You are performing elementary row operations on \( A \), represented on the left-hand side (LHS), and multiplying matrices on the right-hand side (RHS) as part of the row reduction process.
- **Initial \( P \) and \( Q \):**
  - Matrix \( P \) is given as:
    \[
    P = \begin{bmatrix}
    1 & 0 & 0 \\
    0.4 & 0.2 & 1 \\
    -3 & -2 & 1
    \end{bmatrix}
    \]
  - Matrix \( Q \) is given as:
    \[
    Q = \begin{bmatrix}
    1 & -2 & -1 & 0 \\
    0 & 1 & 0 & 0 \\
    0 & 0 & 1 & 0 \\
    0 & 0 & 0 & 1
    \end{bmatrix}
    \]

- **Procedure:** The strategy involves applying elementary row operations on \( A \) and adjusting the product \( PAQ \) to eventually reduce \( A \) to a block matrix, potentially \( I_3 \oplus 0 \), where \( I_3 \) is the identity matrix of size 3x3.

If you would like to further explore the next steps in solving for \( P \) and \( Q \), or if you'd like me to verify or explain any specific portion of this process, please let me know!

Would you like more details on matrix transformations, or do you have specific questions regarding this problem?

Here are five potential follow-up questions:
1. What are elementary row and column operations in the context of matrix factorization?
2. How can you determine if a matrix is non-singular?
3. Can any matrix be reduced to its normal form through operations on \( PAQ \)?
4. What is the significance of \( PAQ \) transformations in solving systems of linear equations?
5. How do you check the correctness of \( P \) and \( Q \) after solving for them?

**Tip:** While performing matrix row operations, always double-check for arithmetic errors in each step to ensure the final form is accurate.

Find the non-singular matrices P and Q such that PAQ is the normal form where A = [3, 2, 4, 5; 5, 1, 4, -2; 1, -4, 11, -14].

This problem involves finding non-singular matrices P and Q such that the matrix product PAQ results in a normal form. The matrix A provided is a 3x4 matrix. The solution involves elementary row and column operations, and adjusting the matrices P and Q accordingly.

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