https://file.aiquickdraw.com/mathbot/file/1727882734597tjpept6x.jpg

To find the rank of the given matrix \( A \), we first need to determine how many linearly independent rows or columns the matrix has. Let's proceed step by step using Gaussian elimination to convert the matrix into row echelon form.

### Matrix \( A \)
\[
A = \begin{bmatrix}
1 & 2 & 3 \\
2 & 3 & 4 \\
3 & 5 & 7
\end{bmatrix}
\]

### Step 1: Form the Augmented Matrix
Start by writing down the matrix in its initial form. We will perform row operations to convert it to row echelon form.

### Step 2: Row Operations
1. **Row 2 Transformation**: To make the element below the first pivot (in row 2, column 1) equal to zero, we can use the operation:
   \[
   R_2 \rightarrow R_2 - 2R_1
   \]
   This yields:
   \[
   A = \begin{bmatrix}
   1 & 2 & 3 \\
   0 & -1 & -2 \\
   3 & 5 & 7
   \end{bmatrix}
   \]

2. **Row 3 Transformation**: To make the element below the first pivot (in row 3, column 1) equal to zero, we use:
   \[
   R_3 \rightarrow R_3 - 3R_1
   \]
   This gives:
   \[
   A = \begin{bmatrix}
   1 & 2 & 3 \\
   0 & -1 & -2 \\
   0 & -1 & -2
   \end{bmatrix}
   \]

3. **Row 3 Simplification**: To make row 3 simpler, notice that rows 2 and 3 are now identical. Therefore:
   \[
   R_3 \rightarrow R_3 - R_2
   \]
   This makes row 3 all zeros:
   \[
   A = \begin{bmatrix}
   1 & 2 & 3 \\
   0 & -1 & -2 \\
   0 & 0 & 0
   \end{bmatrix}
   \]

### Step 3: Determine the Rank
The matrix is now in row echelon form. We have two non-zero rows, which means the rank of the matrix is **2**.

### Summary
- The rank of the matrix \( A \) is **2**.

Would you like more details about rank determination or further assistance?

Here are some related questions that might interest you:
1. What is the definition of matrix rank in general?
2. How do we determine if rows or columns of a matrix are linearly independent?
3. What are some practical applications of finding the rank of a matrix?
4. How does Gaussian elimination differ from Gauss-Jordan elimination?
5. What is the significance of the rank in relation to the solutions of a system of linear equations?

**Tip:** The rank of a matrix is a measure of its non-degeneracy—it tells you the maximum number of linearly independent rows or columns in the matrix.

Find the rank of the matrix A = [[1, 2, 3], [2, 3, 4], [3, 5, 7]]

Determine the rank of the matrix A = [[1, 2, 3], [2, 3, 4], [3, 5, 7]] using Gaussian elimination. This step-by-step guide walks you through the process of converting the matrix to row echelon form to find the rank. Appropriate for students at the advanced high school or college level.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation