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reduce the matrix into normal form and find its rank

The given matrix is:

\[
\begin{pmatrix}
1 & 3 & 6 & -1 \\
1 & 4 & 5 & 1 \\
1 & 5 & 4 & 3
\end{pmatrix}
\]

After performing row reduction (Gaussian elimination), the matrix in row echelon form is:

\[
\begin{pmatrix}
1 & 3 & 6 & -1 \\
0 & 1 & -1 & 2 \\
0 & 0 & 0 & 0
\end{pmatrix}
\]

The rank of the matrix is \(2\) since there are two non-zero rows in the row-reduced form.

Would you like more details on how this reduction was performed or have any further questions? Here are some related questions:

1. How is the rank of a matrix defined?
2. Can a matrix have a rank greater than its number of rows or columns?
3. What is the significance of row echelon form in solving linear equations?
4. How can we find the null space of this matrix?
5. How does Gaussian elimination work in practice?

**Tip**: The rank of a matrix helps in determining the number of linearly independent rows or columns in the matrix.

Reduce the matrix into normal form and find its rank.

Learn how to reduce a matrix to row echelon form using Gaussian elimination and determine its rank. This guide covers core concepts of linear algebra, including matrix theory and the rank-nullity theorem. Suitable for undergraduate mathematics students.

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