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CHAPTER 8.1. Matrices and Systems of Equations. Matrix - a streamlined technique for solving systems of linear equations that involves the use of a rectangular array of numbers. M rows. N columns. M x N. Order of Matrices. system. augmented matrix. coefficient matrix.

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chapter 8 1

CHAPTER 8.1

Matrices and Systems of Equations

slide2
Matrix- a streamlined technique for solving systems of linear equations that involves the use of a rectangular array of numbers

M rows

N columns

M x N

slide4

system

augmentedmatrix

coefficientmatrix

slide5

Writing an Augmented Matrix

1. Begin by writing the linear system and aligning the variables.

2. Next, use the coefficients and constant terms as the matrix entries. Include zeroes for each missing coefficients.

slide6

Elementary Row Operations

1. Interchange two rows.

2. Multiply a row by a nonzero constant.

3. Add a multiple of a row to another row.

slide7

Associated Augmented Matrix

Use back-substitution to find the solution

slide8

Row-Echelon Form and Reduced Row-Echelon Form

1. All rows consisting of zeroes occur at the bottom of the matrix.

2. For each row that does not consist of zeroes, the first nonzero entry is 1(called a leading 1).

3. For two successive (nonzero) rows, the leading 1 in the higher row is farther to the left than the leading 1 in the lower row.

Reduced row-echelon form- if every column that has a leading 1 has zeroes in every position above and below its leading 1.

slide9

For example:

Row-Echelon Form

Reduced row-echelon form

slide10

Solve the system

Switch row 1 with row 2

slide12

Gaussian Elimination with Back-Substitution

1. Write the augmented matrix of the system of linear equations.

2. Use elementary row operations to rewrite the augmented matrix in row-echelon form.

3. Write the system of linear equations corresponding to the matrix in row-echelon form and use back-substitution to find the solution.

slide13

In row 3,

Solve the system

inconsistent, no solution

slide15

Let

where a is

A real number, then the solution set has the form

=infinite number of solutions

Solve the system