1 1 -3 7 9 -9
1 2 -4 10 13 -12
1 -1 -1 1 1 -3
1 -3 1 -5 -7 3
1 -2 0 0 -5 -4 Find the rank

To determine the rank of a matrix, we need to find the number of linearly independent rows or columns. The rank is equivalent to the maximum number of linearly independent vectors that span the row or column space of the matrix. This can be done by reducing the matrix to its **row echelon form** (or reduced row echelon form) using Gaussian elimination.

The matrix you've provided is:

\[
\begin{bmatrix}
1 & 1 & -3 & 7 & 9 & -9 \\
1 & 2 & -4 & 10 & 13 & -12 \\
1 & -1 & -1 & 1 & 1 & -3 \\
1 & -3 & 1 & -5 & -7 & 3 \\
1 & -2 & 0 & 0 & -5 & -4 \\
\end{bmatrix}
\]

Let's perform row reduction to find its rank.

### Step 1: Subtract row 1 from all other rows
After performing row operations, we will assess the number of non-zero rows.After performing row reduction, the matrix in row echelon form is:

\[
\begin{bmatrix}
1 & 0 & -2 & 0 & 9 & 2 \\
0 & 1 & -1 & 0 & 7 & 3 \\
0 & 0 & 0 & 1 & -1 & -2 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
\end{bmatrix}
\]

This matrix has three non-zero rows. Therefore, the **rank** of the matrix is:

\[
\text{Rank} = 3
\]

Would you like more details about the row-reduction process?

Here are some related questions to deepen your understanding:
1. What is the significance of matrix rank in linear algebra?
2. How does rank help in solving systems of linear equations?
3. What are the implications if a matrix has full rank?
4. How does row echelon form differ from reduced row echelon form?
5. Can rank be determined by examining column vectors instead of rows?

**Tip:** A matrix is said to have full rank if its rank is equal to the smallest dimension (number of rows or columns).

Find the rank of the matrix: 
1 1 -3 7 9 -9
1 2 -4 10 13 -12
1 -1 -1 1 1 -3
1 -3 1 -5 -7 3
1 -2 0 0 -5 -4

Learn how to find the rank of a matrix using Gaussian elimination. This example involves reducing a 5x6 matrix to its row echelon form to determine the number of linearly independent rows. Suitable for advanced high school or college-level linear algebra courses.

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