https://file.aiquickdraw.com/mathbot/file/17271917124220u3l98kh.jpg

The image shows several matrices and their corresponding ranks calculated through row reduction to normal form. Let's analyze each matrix and the result:

1. **Matrix (1):**
   \[
   \begin{pmatrix}
   2 & -4 & 3 & -10 & 0 \\
   1 & -2 & -1 & -4 & 2 \\
   0 & 1 & -1 & 3 & 1 \\
   4 & -7 & 4 & -4 & 5
   \end{pmatrix}
   \]
   The rank of this matrix is \( \text{Rank}(A_1) = 4 \).

2. **Matrix (2):**
   \[
   \begin{pmatrix}
   2 & 1 & 2 & 4 \\
   4 & 2 & 5 & 8 \\
   3 & 1 & 4 & 6 \\
   1 & 1 & 2 & 2
   \end{pmatrix}
   \]
   The rank of this matrix is \( \text{Rank}(A_2) = 3 \).

3. **Matrix (3):**
   \[
   \begin{pmatrix}
   2 & 3 & -1 & -1 \\
   1 & 1 & -2 & -4 \\
   3 & 1 & 3 & 2 \\
   6 & 3 & 0 & -7
   \end{pmatrix}
   \]
   The rank of this matrix is \( \text{Rank}(A_3) = 3 \).

4. **Matrix (4):**
   \[
   \begin{pmatrix}
   0 & 1 & -3 & -4 \\
   1 & 0 & 1 & 1 \\
   3 & 1 & 0 & 2 \\
   1 & 1 & -2 & 0
   \end{pmatrix}
   \]
   The rank of this matrix is \( \text{Rank}(A_4) = 2 \).

5. **Matrix (5):**
   \[
   \begin{pmatrix}
   4 & 3 & 2 & 1 \\
   5 & 1 & -1 & -2 \\
   0 & 1 & 2 & 3 \\
   1 & -1 & 3 & -2
   \end{pmatrix}
   \]
   The rank of this matrix is \( \text{Rank}(A_5) = 4 \).

6. **Matrix (6):**
   \[
   \begin{pmatrix}
   1 & 2 & 1 & 0 \\
   -2 & 4 & 3 & 0 \\
   1 & 0 & 2 & -1
   \end{pmatrix}
   \]
   The rank of this matrix is \( \text{Rank}(A_6) = 3 \).

---

Let me know if you'd like further explanation on how to compute the ranks of these matrices, or if you want to go over the row reduction steps!

Here are five follow-up questions you might consider:
1. What is the method for reducing a matrix to its row echelon form?
2. How does row echelon form help in determining the rank of a matrix?
3. What is the difference between row echelon form and reduced row echelon form?
4. How is the rank of a matrix related to the linear independence of its rows or columns?
5. Can a matrix have different row and column ranks?

**Tip:** The rank of a matrix represents the maximum number of linearly independent row or column vectors within the matrix.

Find the rank of the matrix by reducing to the normal form.

Learn how to calculate the rank of a matrix by reducing it to row echelon form. This example includes step-by-step matrix reduction and highlights key concepts like matrix rank, Gaussian elimination, and linear independence.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation