https://file.aiquickdraw.com/mathbot/file/1726463788899vvw95rgl.jpg

The given problem asks to reduce the matrix \( A \) to its normal form (Row Echelon Form) and find its rank. The matrix \( A \) provided in the image is:

\[
A = \begin{bmatrix} 1 & 2 & -1 & 4 \\ 2 & 4 & 3 & 4 \\ 1 & 2 & 3 & 4 \\ -1 & -2 & 6 & -7 \end{bmatrix}
\]

Let's proceed with the following steps:

1. **Row Reduction:** Use elementary row operations to convert this matrix into Row Echelon Form (REF).
2. **Rank Determination:** The rank of the matrix is equal to the number of non-zero rows in the Row Echelon Form.

Let's start by performing the row reduction and finding the rank. I'll handle this computation now.The matrix \( A \) has been reduced to its Row Echelon Form:

\[
\begin{bmatrix} 
1 & 2 & 0 & 0 \\ 
0 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 1 \\ 
0 & 0 & 0 & 0 
\end{bmatrix}
\]

The rank of this matrix is 3, as there are 3 non-zero rows in the Row Echelon Form.

Would you like more details on the row reduction steps, or do you have any questions?

Here are 5 related questions that might interest you:
1. How do elementary row operations affect the rank of a matrix?
2. Can the rank of a matrix change if columns or rows are swapped?
3. What is the difference between Row Echelon Form and Reduced Row Echelon Form?
4. How is the rank of a matrix related to its solutions in a system of linear equations?
5. What properties can be deduced about a matrix if its rank is equal to its number of columns?

**Tip:** The rank of a matrix gives insight into the linear independence of its rows or columns, which is crucial in solving linear systems and understanding the matrix's properties.

Reduce the matrix to normal form and find the rank of the matrix A:
A = [ [1, 2, -1, 4], [2, 4, 3, 4], [1, 2, 3, 4], [-1, -2, 6, -7] ]

Solve for the normal form and find the rank of the matrix A using row reduction. This step-by-step guide explains the process of reducing matrix A to Row Echelon Form and determining its rank. Key concepts include elementary row operations and the matrix rank theorem, suitable for college-level students studying linear algebra.

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