7.3 & 7.4 – MATRICES AND SYSTEMS OF

EQUATIONS

IN THIS SECTION, YOU WILL LEARN TO

Write a matrix and identify its order Perform elementary row operations

on matrices Use matrices and Gaussian

elimination to solve systems of linear equations

Use matrices and Gauss-Jordan elimination to solve systems of linear equations

DEFINITION OF A MATRIX:

11 12 13 1

21 22 23 2

31 32 33 3

1 2 3

...

...

...

. . . .

. . . .

...

n

n

n

m m m mn

n columns

a a a a

a a a a

a a a am rows

a a a a

DEFINITION OF A MATRIX:

) means the entry of the number in the row

and the column

thij

th

a a i

j

c) If , then the matrix is called a square

matrix

m n

11 22 33d) In the square matrix, the entries , , ..

are called the main diagonal entries.

a a a

b) An matrix has rows (horizontal lines)

and columns (vertical lines)

mxn m

n

ORDER OF THE MATRIX:

1)

2)

3)

4)

4 : 1 1 matrix

0 4 3 1 : 1 4 matrix

0 4 3 1:

1 3 0 7

2 4 matrix

0 4 3

1 3 0 :

5 1 4

3 3 matrix

FORMS OF A MATRIX:

1) A matrix is derived from a system of linear equations which can be represented in a coefficient matrix or an augmented matrix form:

FORMS OF A MATRIX:

a) System of linear equations:

b) Coefficient Form: c) Augmented Form:

4 3 4 1

3 3 33

4 22

x y z

x y z

x y

4 3 4

3 1 3

1 4 0

4 3 4 1

3 1 3 33

1 4 0 22

ELEMENTARY ROW OPERATIONS:

1) These operations are used to solve for the values in the system of linear equations.

a) Interchange two rows b) Multiply a row by a nonzero constant c) Add a multiple of a row to another

row

MATRIX EXAMPLES:

Solve:Linear Method Matrix Method

* We are going to solve this problem using the linear system method and the matrix elementary row operation method side by side.

2 3 9

3 4

2 5 5 17

x y z

x y

x y z

1 2 3 9

1 3 0 4

2 5 5 17

STEP 1:

Add the first row to the second row:

LINEAR METHOD MATRIX METHOD

1 2

1 2 3 9

0 1 3 5

2 5 5 17

R R

Add the first equation

to the second equation:

2 3 9

3 5

2 5 5 17

x y z

y z

x y z

STEP 2:

Add times the first row to the third row:

Add times the first equation to

the third equation:

2 2

2 3 9

3 5

1

x y z

y z

y z

1 2 3 9

0 1 3 5

0 1 1 1

LINEAR METHOD MATRIX METHOD

1 32R R

STEP 3:

Add the second row to the

third row:

Add the second equation to the third equation:

2 3 9

3 5

2 4

x y z

y z

z

1 2 3 9

0 1 3 5

0 0 2 4

MATRIX METHODLINEAR METHOD

2 3R R

STEP 4:

Multiply the third row by :

Multiply the thirdequation by :

2 3 9

3 5

2

x y z

y z

z

1 2 3 9

0 1 3 5

0 0 1 2

MATRIX METHODLINEAR METHOD

1

2

1

2

3

1

2R

STEP 5:

Use back substitution to solve for the remaining variables.

3 5

3 2 5

6 5

1

y z

y

y

y

2 3 9

2 1 3 2 9

2 6 9

1

x y z

x

x

x

Therefore, 1, 1 and 2 1, 1,2x y z

ROW ECHELON AND REDUCED ROW ECHELON FORMS:

A matrix in row-echelon form has the following properties:a) All rows entirely of zeros is at the bottom of the matrix.b) For any rows not made entirely of zeros, the first nonzero entry is 1. (leading 1)c) For two successive rows, the leading 1 in the higher row is farther to the left than the lower row.

ROW ECHELON AND REDUCED ROW ECHELON FORMS:

Row Echelon Form:

Not in Row Echelon Form:

1 0 2

0 1 5

0 0 0

0 1 2

0 0 0

1 3 0

ROW ECHELON AND REDUCED ROW ECHELON FORMS: Reduced Row Echelon Form: a matrix is

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

 

Not in Row Echelon Form:

1 0 0 2

0 1 0 3 2, 3, 5

0 0 1 5

x y z

GAUSSIAN ELIMINATION WITH BACK SUBSTITUTION: (NAMED AFTER CARL FRIEDRICH GAUSS)

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.

EXAMPLE #1:

Solve: 2 3

2 6

3 4 3 4

x y z

x z

x y z

2 1 1 3

1 0 2 6

3 4 3 4

EXAMPLE #1:

EXAMPLE #1:

GAUSS-JORDAN ELIMINATION:

A second method of elimination is named after Carl Friedrich Gauss and Wilhelm Jordan. You continue with the reduction process until a reduced row-echelon form is obtained.

Solve using this method.

2 3

2 6

3 4 3 4

x y z

x z

x y z

GAUSS-JORDAN ELIMINATION EXAMPLE:

GAUSS-JORDAN ELIMINATION EXAMPLE:

SYSTEM WITH NO SOLUTION:

It is possible for a system of linear equations to have no solution. If you obtain a row with zeros except for the last entry, it is unnecessary to continue the elimination process. The system has no solution, or is inconsistent.

Example:

1 0 2 6

0 0 0 9 No Solution

0 0 1 2

SYSTEM WITH INFINITE MANY SOLUTION:

It is possible for a system of linear equations to have an infinite number of solutions.

Example: 2 3

2 6

x y z

x z

SYSTEM WITH INFINITE MANY SOLUTION:

Example #1:

2 1

2 1 1 3 2 1 1 32

1 0 2 6 0 1 5 9R R

2 1

2 0 4 12

0 1 5 9R R

SYSTEM WITH INFINITE MANY SOLUTION:

2 4 12

5 9

Solving for and in terms of

2 4 12 5 9

2 6 5 9

Substituting as any real number :

2 6 5 9

The solution set has the form : 2 6, 5 9,

x z

y z

x y z

x z y z

x z y z

a

x a y a z a

a a a