https://file.aiquickdraw.com/mathbot/file/1726282038309wkeppmih.jpg

To find the rank of the given matrix, let's first identify the matrix from the image:

\[
A = \begin{pmatrix}
2 & 3 & -2 & 4 \\
3 & -2 & 1 & 2 \\
3 & 2 & 3 & 4 \\
-2 & 4 & 0 & 5
\end{pmatrix}
\]

I will now compute the rank of this \(4 \times 4\) matrix by transforming it into row echelon form. Let me proceed with the calculation.The rank of the given matrix is \( 4 \). This means that all rows of the matrix are linearly independent.

Would you like more details about the steps, or have any further questions?

Here are five related questions to expand on this topic:
1. What does the rank of a matrix represent in terms of its column and row spaces?
2. How do we perform row reduction to find the rank manually?
3. What is the significance of a full-rank matrix in linear algebra?
4. How can we determine the rank of a matrix using determinants?
5. How does the rank of a matrix relate to the solutions of a system of linear equations?

**Tip:** A square matrix with full rank (rank equal to its size) is invertible, meaning it has a unique inverse.

This problem involves finding the rank of a 4x4 matrix. Learn the step-by-step process of transforming the matrix into row echelon form to determine its rank. Explore key linear algebra concepts like row reduction and the significance of matrix rank.

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