MTH 309 2. Matrices and elementary row operations Next ...

MTH 309

2. Matrices and elementary row operations

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How to solve a system of linear equations

system of equations - 1 + 2 2 + 3 3 = 4

24

1+6 3=9 1- 2-3 3

=0

make a matrix

augmented matrix

solutions 1 =

2 3

= =

read off solutions

Gauss-Jordan elimination

matrix in reduced row echelon form

5

Matrices matrix = rectangular array of numbers

Example.

123 456

1 20

7 8

-5 10

1 7

6 43

Note Every system of linear equations can be represented by a matrix.

Example.

- 1 + 2 2 + 3 3 = 4

24

1 1

+6 3=9 - 2-3 3

=

0

6

Elementary row operations:

1) Interchange of two rows.

Example.

1234

0 1 5 1

4307

2) Multiplication of a row by a non-zero number.

Example.

1234

0 1 5 1

4307

3) Addition of a multiple of one row to another row.

Example.

1234

0 1 5 1

4307

7

Proposition

Elementary row operations do not change solutions of the system of equations represented by a matrix.

system of linear equations

different systems same solutions

system of linear equations

augmented matrix

elementary row operation

augmented matrix

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