Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

a. Using technology, generate a random $4 \times 3$ matrix A. (The entries may be either single-digit integers or numbers between 0 and 1, depending on the technology you are using.) Find $rref (A)$ . Repeat this experiment a few times.
b. What does the reduced row-echelon form of most $4 \times 3$ matrices look like? Explain.

a. A $3 \times 4$ matrix $A$ has $r r e f (A) = [\begin{matrix} 1 & 0 & - 1 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]$

b. The reduced row-echelon form of matrix has leading 1’s with zero elements below the leading 1’s and non-zero elements above the leading 1’s.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Consider the matrix.

The reduced row-echelon form of a matrix has number of leading 1’s in each row and is denoted as $r r e f (A) = [\begin{matrix} 1 & \dots & a \\ 0 & ⋱ & ⋮ \\ 0 & 0 & 1 \end{matrix}]$ .

The matrix is,

$A = [\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ 1 & 2 & 3 \end{matrix}]$

Step 2: Find the reduced row-echelon form.

Consider the matrix

$A = [\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ 1 & 2 & 3 \end{matrix}]$

$= A = [\begin{matrix} 7 & 8 & 9 \\ 0 & \frac{6}{7} & \frac{12}{7} \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]$

$r r e f (A) [\begin{matrix} 1 & 0 & - 1 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]$

localid="1659343693826"

T h e r e f o r e, r r e f (A) [\begin{matrix} 1 & 0 & - 1 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]

Step 3: Explanation for the reduced row-echelon form.

A matrix in reduced row echelon form is used to solve systems of linear equations. There are four prerequisites for the reduced row echelon form:

The number 1 is the first non-zero integer in the first row (the leading entry).
The second row begins with the number 1, which is more to the right than the first row's leading item. The number 1 must be further to the right in each consecutive row.
Each row's first item must be the sole non-zero number in its column.
Any rows that are not zero are pushed to the bottom of the matrix.

Consider the reduced row-echelon form.

$r r e f (A) [\begin{matrix} 1 & 0 & - 1 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]$

The reduced row-echelon form of matrix has leading 1’s with zero elements below the leading 1’s and non-zero elements above the leading 1’s.

Step 4: Final answer.

a. $r r e f (A) [\begin{matrix} 1 & 0 & - 1 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]$

b. The reduced row-echelon form of matrix has leading 1’s with zero elements below the leading 1’s and non-zero elements above the leading 1’s.

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Linear Algebra With Applications

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Short Answer

Step by Step Solution

Step 1: Consider the matrix.

Step 2: Find the reduced row-echelon form.

Step 3: Explanation for the reduced row-echelon form.

Step 4: Final answer.

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Linear Algebra With Applications

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Short Answer

Step by Step Solution

Step 1: Consider the matrix.

Step 2: Find the reduced row-echelon form.

Step 3: Explanation for the reduced row-echelon form.

Step 4: Final answer.

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