https://file.aiquickdraw.com/mathbot/file/1726192235269s0nxvrk1.jpg

To find the rank of the matrix in normal form, we'll first apply row reduction (Gaussian elimination) to bring the matrix into its row echelon form (REF) or reduced row echelon form (RREF). The rank of the matrix will be the number of non-zero rows in this form.

Here is the matrix provided in the image:

\[
\begin{pmatrix}
1 & 3 & 2 & 1 \\
1 & 2 & 3 & 2 \\
3 & 1 & 1 & 3 \\
0 & 1 & 0 & 0
\end{pmatrix}
\]

Let me perform the row reduction process and find the rank for you.The matrix has been reduced to the following row echelon form:

\[
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}
\]

This is the identity matrix, indicating that all rows are independent. Therefore, the **rank of the matrix** is **4**.

Would you like further details or have any additional questions?

Here are 5 related questions for you to explore:
1. What is the significance of a matrix rank in solving linear systems?
2. How does row reduction help determine the rank of a matrix?
3. Can a matrix have a rank greater than the number of its rows or columns?
4. What is the difference between row echelon form (REF) and reduced row echelon form (RREF)?
5. How does matrix rank relate to the concept of linear independence?

**Tip:** The rank of a matrix can also

Discover how to determine the rank of a matrix by converting it to normal form through Gaussian elimination. This advanced topic in linear algebra covers key concepts such as matrix theory and the rank theorem, tailored for university-level students.

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