Find the rank of the matrix
 1 -3 4 -1 9
 -2 6 -6 -1 -10
 -3 9 -6 -6 -3 
  3 -3 4 9 0

We can find the rank of a matrix by performing row reduction (Gaussian elimination) to bring it to row echelon form, then counting the number of non-zero rows. The given matrix is:

\[
\begin{bmatrix}
1 & -3 & 4 & -1 & 9 \\
-2 & 6 & -6 & -1 & -10 \\
-3 & 9 & -6 & -6 & -3 \\
3 & -3 & 4 & 9 & 0
\end{bmatrix}
\]

Let's perform Gaussian elimination to find the rank.

1. We start by using the first row to eliminate elements in the first column of the other rows.
2. Then, we proceed similarly for the second column, and so on, reducing the matrix to row echelon form.
  
I'll do the calculations to simplify this matrix.The rank of the given matrix is 3.

Would you like any further details on the steps for Gaussian elimination, or do you have any other questions?

Here are 5 related questions to expand on this topic:
1. How is Gaussian elimination performed step by step?
2. What is the relationship between rank and the number of pivots in a matrix?
3. Can the rank of a matrix change if we transpose it?
4. How does the rank of a matrix relate to its determinant?
5. What does the rank tell us about the solutions to a system of linear equations?

**Tip:** The rank of a matrix also indicates the maximum number of linearly independent row (or column) vectors in the matrix.

Find the rank of the matrix
1 -3 4 -1 9
-2 6 -6 -1 -10
-3 9 -6 -6 -3
3 -3 4 9 0

Learn how to find the rank of a 4x5 matrix using Gaussian elimination. This problem focuses on matrix theory and involves reducing the matrix to row echelon form. Suitable for high school and college students studying linear algebra.

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