Math Problem Statement

Reduce A into Echelon form and find its rank, where A = [[3, 4, 5, 6, 7], [4, 5, 6, 7, 8], [5, 6, 7, 8, 9], [10, 11, 12, 13, 14], [15, 16, 17, 18, 19]]

Solution

To reduce the matrix AA to echelon form and find its rank, let's proceed step-by-step.

The matrix AA is given as:

A=(34567456785678910111213141516171819)A = \begin{pmatrix} 3 & 4 & 5 & 6 & 7 \\ 4 & 5 & 6 & 7 & 8 \\ 5 & 6 & 7 & 8 & 9 \\ 10 & 11 & 12 & 13 & 14 \\ 15 & 16 & 17 & 18 & 19 \end{pmatrix}

Step 1: Apply Row Operations to Reach Echelon Form

The goal of reducing a matrix to row echelon form is to create leading 1s in each row and zeros below each leading 1. Let's apply the row operations to simplify this matrix.

  1. Start with row 1 as the pivot row and create zeros below the first element (3 in the first row).

    • R2→R2−43R1R_2 \rightarrow R_2 - \frac{4}{3} R_1
    • R3→R3−53R1R_3 \rightarrow R_3 - \frac{5}{3} R_1
    • R4→R4−103R1R_4 \rightarrow R_4 - \frac{10}{3} R_1
    • R5→R5−5R1R_5 \rightarrow R_5 - 5 R_1

After performing these operations, we'll continue similarly to clear elements below each pivot in subsequent rows.

Step 2: Continue Row Operations

I’ll carry out these steps computationally and display the reduced echelon form of the matrix along with its rank.import numpy as np from sympy import Matrix

Define the matrix A

A = np.array([ [3, 4, 5, 6, 7], [4, 5, 6, 7, 8], [5, 6, 7, 8, 9], [10, 11, 12, 13, 14], [15, 16, 17, 18, 19] ])

Convert to a sympy Matrix to use row reduction and determine the rank

sympy_matrix = Matrix(A) echelon_form = sympy_matrix.rref()

Extract the echelon form and rank

echelon_matrix, pivot_columns = echelon_form rank = len(pivot_columns) # Rank is the number of pivot columns

echelon_matrix, rank(Matrix([ [1, 0, -1, -2, -3], [0, 1, 2, 3, 4], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0]]), 2)(Matrix([ [1, 0, -1, -2, -3], [0, 1, 2, 3, 4], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0]]), 2)successThe row echelon form of the matrix AA is:

1 & 0 & -1 & -2 & -3 \\ 0 & 1 & 2 & 3 & 4 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix}$$ The rank of matrix $$A$$ is $$2$$, as there are two pivot columns. Would you like further details on each row operation or have any questions? Here are some related questions you might explore: 1. What does the rank of a matrix represent in terms of its columns? 2. How does row echelon form differ from reduced row echelon form? 3. What are the implications of having a rank less than the number of columns? 4. How would you use the rank to determine if the matrix is invertible? 5. Can a matrix have a rank higher than the number of rows? **Tip:** In echelon form, each row with a pivot represents a linearly independent equation or vector.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Matrix Operations
Row Echelon Form
Matrix Rank

Formulas

Row reduction to Echelon form
Rank determination based on pivot columns

Theorems

Rank theorem
Echelon form properties

Suitable Grade Level

Undergraduate Level

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