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find the non singular matrices P and Q such that paq is of the normal form and find its rank

To find nonsingular matrices \( P \) and \( Q \) such that \( PAQ \) is in normal form, and to determine the rank of the matrix, we can approach this by reducing the matrix to its row echelon form or diagonal form through row and column operations. Here's how we can proceed:

### Step 1: Identify the given matrix \( A \)

From the image, the matrix \( A \) appears to be:
\[
A = \begin{pmatrix}
2 & -3 & 1 \\
2 & -3 & 1 \\
1 & 1 & 1
\end{pmatrix}
\]

### Step 2: Perform elementary row and column operations to reduce \( A \) to normal form (RREF or diagonal form)

Let's first check the rank of the matrix by row reducing it to an echelon form.

### Step 3: Calculate the rank

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Here are some related questions to think about:
1. What is the process of finding the row echelon form of a matrix?
2. How do elementary row operations affect the rank of a matrix?
3. What is the significance of nonsingular matrices in the context of matrix transformations?
4. How does the rank of a matrix relate to its invertibility?
5. Can the rank of a matrix change after multiplying by nonsingular matrices?

**Tip:** In matrix theory, the rank of a matrix tells you the maximum number of linearly independent rows or columns.

Find the nonsingular matrices P and Q such that PAQ is of the normal form and find its rank.

Explore how to find nonsingular matrices P and Q such that PAQ is in normal form, and calculate the rank of the matrix. This detailed step-by-step guide covers key linear algebra concepts such as row echelon form, elementary operations, and matrix rank. Suitable for university-level students.

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