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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 1: Q1.2 28E (page 19)

Consider an n脳mmatrix A. Can you transform rref(A)into Aby a sequence of elementary row operations? See Exercise 27.

Short Answer

Expert verified

Yes, for transforming the matrix A into rref we can do the sequence of operations.

Step by step solution

01

Row operations

In matrix there are three types of row operations for transforming the matrix of the form n脳m:

Swapping the rows: In this operation, we can swap any row with any other row.
Constant multiplications with row: In this operation, we can multiply any non-zero scalar with any row.
Subtracting or adding the row: In this operation, we can subtract or add another row.

02

Example for explanation

Suppose a matrix A of order 2脳2, $A = [\begin{matrix} 1 & 3 \\ 3 & 6 \end{matrix}]$

We will use the sequence of row operations for transforming the equation to rref.

$A = [\begin{matrix} 1 & 3 \\ 3 & 6 \end{matrix}] R_{2} 鈫� R_{2} - 3 R_{1} ~ [\begin{matrix} 1 & 3 \\ 0 & - 3 \end{matrix}] R_{2} 鈫� \frac{1}{- 3} R_{2} ~ [\begin{matrix} 1 & 3 \\ 0 & 1 \end{matrix}] R_{1} 鈫� R_{1} - 3 R_{2} ~ [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$

Yes, for transforming the matrix A into rref we can do the sequence of operations.

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