Concept Byte: Matrices and Solving Systems

Use With Lesson 6-3

EXTENSION

A matrix is a rectangular arrangement of numbers in rows and columns. The plural of matrix is matrices. You will learn more about matrix operations, including adding and subtracting matrices, in Chapter 12.

You can use a special type of matrix, called an augmented matrix, to solve a system of linear equations. An augmented matrix is formed using the coefficients and constants in the equations in a system. The equations must be written in standard form.

Recall the operations you performed when you solved systems using elimination. You can perform similar operations on the rows of an augmented matrix.

You can perform any of the following row operations on an augmented matrix to produce an equivalent augmented matrix.

Interchange two rows. [ 7 6 4 5 | 10 − 5 ] → [ 4 5 7 6 | − 5 10 ] . matrix with 1 row and 2 columns , row1 column 1 , table with 2 rows and 2 columns , row1 column 1 , 7 , column 2 6 , row2 column 1 , 4 , column 2 5 , end table . vertical line , column 2 table with 2 rows and 1 column , row1 column 1 , 10 , row2 column 1 , negative 5 , end table , end matrix . rightwards arrow . matrix with 1 row and 2 columns , row1 column 1 , table with 2 rows and 2 columns , row1 column 1 , 4 , column 2 5 , row2 column 1 , 7 , column 2 6 , end table . vertical line , column 2 table with 2 rows and 1 column , row1 column 1 , negative 5 , row2 column 1 , 10 , end table , end matrix

Multiply a row by any constant except 0. [ 7 6 4 5 | 10 − 5 ] → [ 7 6 2 ( 4 ) 2 ( 5 ) | 10 2 ( − 5 ) ] → [ 7 6 8 10 | 10 − 10 ] . matrix with 1 row and 2 columns , row1 column 1 , table with 2 rows and 2 columns , row1 column 1 , 7 , column 2 6 , row2 column 1 , 4 , column 2 5 , end table . vertical line , column 2 table with 2 rows and 1 column , row1 column 1 , 10 , row2 column 1 , negative 5 , end table , end matrix . rightwards arrow . matrix with 1 row and 2 columns , row1 column 1 , table with 2 rows and 2 columns , row1 column 1 , 7 , column 2 6 , row2 column 1 , 2 open 4 close , column 2 2 open 5 close , end table . vertical line , column 2 table with 2 rows and 1 column , row1 column 1 , 10 , row2 column 1 , 2 open negative 5 close , end table , end matrix . rightwards arrow . matrix with 1 row and 2 columns , row1 column 1 , table with 2 rows and 2 columns , row1 column 1 , 7 , column 2 6 , row2 column 1 , 8 , column 2 10 , end table . vertical line , column 2 table with 2 rows and 1 column , row1 column 1 , 10 , row2 column 1 , negative 10 , end table , end matrix

Add a multiple of one row to another row. [ 7 6 4 5 | 10 − 5 ] → [ 7 + 2 ( 4 ) 6 + 2 ( 5 ) 4 5 | 10 + 2 ( − 5 ) − 5 ] → [ 15 16 4 5 | 0 − 5 ] . matrix with 1 row and 2 columns , row1 column 1 , table with 2 rows and 2 columns , row1 column 1 , 7 , column 2 6 , row2 column 1 , 4 , column 2 5 , end table . vertical line , column 2 table with 2 rows and 1 column , row1 column 1 , 10 , row2 column 1 , negative 5 , end table , end matrix . rightwards arrow . matrix with 1 row and 2 columns , row1 column 1 , table with 2 rows and 2 columns , row1 column 1 , 7 plus 2 open 4 close , column 2 6 plus 2 open 5 close , row2 column 1 , 4 , column 2 5 , end table . vertical line , column 2 table with 2 rows and 1 column , row1 column 1 , 10 plus 2 open negative 5 close , row2 column 1 , negative 5 , end table , end matrix . rightwards arrow . matrix with 1 row and 2 columns , row1 column 1 , table with 2 rows and 2 columns , row1 column 1 , 15 , column 2 16 , row2 column 1 , 4 , column 2 5 , end table . vertical line , column 2 table with 2 rows and 1 column , row1 column 1 , 0 , row2 column 1 , negative 5 , end table , end matrix

To solve a system using an augmented matrix, choose row operations that will transform the augmented matrix into a matrix with 1's along the main diagonal (top left to lower right) and 0's above and below the main diagonal, as shown below.

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