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行化简

行化简

1. Introduction

Introduction

2. The Elimination Method

The Elimination Method

3. Echelon Forms

Echelon Forms

4. The Row Reduction Algorithm

The Row Reduction Algorithm

5. Which row operation is legal?

Which of the following is not a legal row operation?

6. Reduced row echelon form

Which of the following matrices is in reduced row echelon form?

7. Back substitution

The augmented matrix

(1−23401250048)\left(\begin{array}{ccc|c} 1 & -2 & 3 & 4 \\ 0 & 1 & 2 & 5 \\ 0 & 0 & 4 & 8 \end{array}\right)

is in row echelon form. Back-substitute and enter the value of xx.

8. Counting pivot columns

The matrix

(1204001−10000)\left(\begin{array}{ccc|c} 1 & 2 & 0 & 4 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{array}\right)

is in reduced row echelon form. How many pivot columns does it have? Enter a number.

9. Solve by row reduction

Solve the system by row reduction and enter the solution as x,y,zx,y,z (for example: 1,0,-2).

{x+y+z=52x−y+z=9x+2y−2z=−8\left\{\begin{aligned} x + y + z &= 5 \\ 2x - y + z &= 9 \\ x + 2y - 2z &= -8 \end{aligned}\right.

10. What row reduction does

Row reducing the augmented matrix of a linear system is useful because the row echelon form reached has which property?

11. One elimination step

In the augmented matrix

(123259)\left(\begin{array}{cc|c} 1 & 2 & 3 \\ 2 & 5 & 9 \end{array}\right)

replace row 2 by row 2 minus 2 times row 1, and enter the three entries of the new row 2 as a,b,ca,b,c.

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