Matrices and Determinants

Advanced Math

Chapter 10

Matrices and Systems of Equations

Advanced Math 10.1

Advanced Math 10.1 3

Matrix

• Rectangular array of real numbers• Plural: matrices• An m x n matrix has m rows and n columns• Example: a 3 x 2 matrix

11 12

21 22

31 32

a a

a a

a a

Row 1

Row 2

Row 3

Column 1 Column 2

Advanced Math 10.1 4

Matrix entry

• aij

• Each number in the matrix• Has a double subscript to represent row i

and column j• Example: a23 means the number in the

second row, third column

Advanced Math 10.1 5

Matrix order

• m x n• Number of rows x Number of columns

5 8 2 3

7 12 14 5

22 4 5 2

Order: 3 x 4

2 4

5 9

Order: 2 x 2

Square matrix

12 7 6

19 5 72

15 18 2

6 1 0

Order: 4 x 3

Advanced Math 10.1 6

Row matrix

• Has only 1 row

3 5 2

Advanced Math 10.1 7

Column matrix

• Has only 1 column

3

12

5

8

Advanced Math 10.1 8

Matrices from a system of equations

• Line up columns by variable with constant on the right

• Don’t forget zeros

3 4 2 1

: 4

2 3 2

x y z

system x y z

x z

3 4 2 1

1 1 1 4

2 0 3 2

augmented

matrix

Advanced Math 10.1 9

Reduced row-echelon form

• Any row of all zeros must be at the bottom• For all other rows, the first nonzero entry is

1 (called a leading 1)• 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

• Every column that has a leading 1 has zeros in every position above and below its leading 1.

Advanced Math 10.1 10

Solving systems of equations

• Enter augmented matrix into calculator• Use rref to find reduced row-echelon form

3 4 2 1

: 4

2 3 2

x y z

system x y z

x z

3 4 2 1

1 1 1 4

2 0 3 2

augmented

matrix

1 0 0 47

0 1 0 19

0 0 1 32

rref

matrix

47

19

32

47,19, 32

x

y

z

Advanced Math 10.1 11

You try

• Solve2 3 24

2 14

7 5 6

x y z

y z

x y

3 3 6 6

2 5

5 8 13 7

x y z

x y z

x y z