Math Problem Statement

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] ]

Solution

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

A=[12−1424341234−1−26−7]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 AA has been reduced to its Row Echelon Form:

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.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Linear Algebra
Matrix Operations
Row Echelon Form
Matrix Rank

Formulas

Elementary Row Operations
Rank of a Matrix: The number of non-zero rows in Row Echelon Form

Theorems

The rank-nullity theorem
Gauss-Jordan Elimination

Suitable Grade Level

College Level - Linear Algebra

Related Recommendation

Finding the Normal Form and Rank of a Matrix A

Matrix Reduction and Rank Calculation: Step-by-Step Solution

Matrix Reduction to Normal Form and Finding Rank Using Row and Column Transformations

Find the Normal Form and Rank of a Matrix using Gaussian Elimination

Finding the Rank of a Matrix Using Normal Form - Step-by-Step Guide