Math Problem Statement

Find rank

Find rank

Solution

It seems that your image shows a matrix and steps involved in reducing it through row operations to find its rank.

Let's summarize the process of finding the rank of a matrix:

  1. Forming the matrix: The rank of a matrix is the number of linearly independent rows or columns in it.
  2. Row-reducing the matrix: Using elementary row operations (swapping rows, multiplying rows by a non-zero scalar, and adding multiples of one row to another), you reduce the matrix to its row echelon form or reduced row echelon form.
  3. Count the non-zero rows: The rank of the matrix is the number of non-zero rows after it has been reduced.

Looking at the steps outlined in your image:

  • The matrix has been row-reduced through some elementary row operations.
  • Based on the final matrix in the image, the non-zero rows appear to be clearly marked.

I'll interpret and solve this further for rank computation:

  1. Original matrix: 1 & 3 & -1 & 3 \\ 2 & 4 & 3 & -2 \\ 3 & 6 & 0 & 1 \end{pmatrix}$$
  2. Elementary row operations: It seems that rows are being manipulated, as indicated in the notations R3R1R_3 - R_1, etc.

Once the matrix is row-reduced, count the number of non-zero rows. Based on the reduced matrix:

1 & 3 & -1 & 3 \\ 0 & 0 & 4 & -3 \\ 0 & 0 & 0 & 0 \end{pmatrix}$$ From this, there are **2 non-zero rows**. ### **Conclusion:** The **rank of the matrix is 2**, since there are two non-zero rows after row reduction. Would you like a detailed breakdown of the row operations used in this process? Here are five relative questions based on matrix rank: 1. What are elementary row operations, and how do they affect the rank of a matrix? 2. How do you find the rank of a matrix using Gaussian elimination? 3. Can the rank of a matrix be determined using its columns instead of its rows? 4. How does the rank of a matrix relate to the solutions of a system of linear equations? 5. What is the difference between row echelon form and reduced row echelon form? **Tip**: The rank of a matrix can never exceed the smaller of its row or column count.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Linear Algebra
Matrix Rank
Row Reduction

Formulas

Rank of a matrix is the number of non-zero rows in its row echelon form

Theorems

Rank-Nullity Theorem
Gaussian Elimination

Suitable Grade Level

University level, Linear Algebra (1st or 2nd year)