Int Math 2 Section 8-5

SECTION 8-5Matrices and Determinants

Tue, Apr 12

ESSENTIAL QUESTIONS

How do you find the determinant of a 2X2 matrix?

How do you solve systems of equations using determinants?

Where you’ll see this:

Sports, construction, fitness

Tue, Apr 12

VOCABULARY

1. Square Matrix:

2. Determinant:

3. Cramer’s Rule:

Tue, Apr 12

VOCABULARY

1. Square Matrix: A matrix with the same number of rows and columns

2. Determinant:

3. Cramer’s Rule:

Tue, Apr 12

VOCABULARY

1. Square Matrix: A matrix with the same number of rows and columns

2. Determinant: In a 2X2 matrix

a b

c d

⎡

⎣⎢

⎤

⎦⎥, det A = ad - bc.

3. Cramer’s Rule:

Tue, Apr 12

VOCABULARY

1. Square Matrix: A matrix with the same number of rows and columns

2. Determinant: In a 2X2 matrix

a b

c d

⎡

⎣⎢

⎤

⎦⎥, det A = ad - bc.

3. Cramer’s Rule: A method of using determinants of matrices to solve systems of equations

Tue, Apr 12

m x n: m rows and n columns

MATRIX

Tue, Apr 12

m x n: m rows and n columns

MATRIX

−2 2 3

9 6 −3

⎡

⎣⎢

⎤

⎦⎥

3 −1

5 8

⎡

⎣⎢

⎤

⎦⎥

4 −3 0⎡⎣ ⎤⎦

Tue, Apr 12

m x n: m rows and n columns

MATRIX

−2 2 3

9 6 −3

⎡

⎣⎢

⎤

⎦⎥

3 −1

5 8

⎡

⎣⎢

⎤

⎦⎥

4 −3 0⎡⎣ ⎤⎦

2 X 3

Tue, Apr 12

m x n: m rows and n columns

MATRIX

−2 2 3

9 6 −3

⎡

⎣⎢

⎤

⎦⎥

3 −1

5 8

⎡

⎣⎢

⎤

⎦⎥

4 −3 0⎡⎣ ⎤⎦

2 X 3 2 X 2

Tue, Apr 12

m x n: m rows and n columns

MATRIX

−2 2 3

9 6 −3

⎡

⎣⎢

⎤

⎦⎥

3 −1

5 8

⎡

⎣⎢

⎤

⎦⎥

4 −3 0⎡⎣ ⎤⎦

2 X 3 2 X 2 1 X 3

Tue, Apr 12

DETERMINANT

det A =

a b

c d= ad − bc

Tue, Apr 12

EXAMPLE 1

Find the determinant of

0 4

6 7

⎡

⎣⎢

⎤

⎦⎥.

Tue, Apr 12

EXAMPLE 1

Find the determinant of

0 4

6 7

⎡

⎣⎢

⎤

⎦⎥.

ad − bc

Tue, Apr 12

EXAMPLE 1

Find the determinant of

0 4

6 7

⎡

⎣⎢

⎤

⎦⎥.

ad − bc

= 0(7) − 4(6)

Tue, Apr 12

EXAMPLE 1

Find the determinant of

0 4

6 7

⎡

⎣⎢

⎤

⎦⎥.

ad − bc

= 0(7) − 4(6)

= 0 − 24

Tue, Apr 12

EXAMPLE 1

Find the determinant of

0 4

6 7

⎡

⎣⎢

⎤

⎦⎥.

ad − bc

= 0(7) − 4(6)

= 0 − 24

= −24

Tue, Apr 12

CRAMER’S RULE

Tue, Apr 12

CRAMER’S RULE

1. Make sure equations look like Ax + By = C.

Tue, Apr 12

CRAMER’S RULE

1. Make sure equations look like Ax + By = C.

2. Make a 2X2 determinant matrix: x in 1st column, y in 2nd, call A.

Tue, Apr 12

CRAMER’S RULE

1. Make sure equations look like Ax + By = C.

2. Make a 2X2 determinant matrix: x in 1st column, y in 2nd, call A.

3. Make a new 2X2 determinant matrix: Replace x column with equation answers, call Ax.

Tue, Apr 12

CRAMER’S RULE

1. Make sure equations look like Ax + By = C.

2. Make a 2X2 determinant matrix: x in 1st column, y in 2nd, call A.

3. Make a new 2X2 determinant matrix: Replace x column with equation answers, call Ax.

4. Make another 2X2 determinant matrix: Replace y column with equation answers, call Ay.

Tue, Apr 12

CRAMER’S RULE

Tue, Apr 12

CRAMER’S RULE

5. Divide Ax by A and Ay by A.

Tue, Apr 12

CRAMER’S RULE

5. Divide Ax by A and Ay by A.

6. Check answer and rewrite solution.

Tue, Apr 12

EXAMPLE 2Solve the system of equations using Cramer’s rule (matrices).

3x − 7 y = −6

x + 2 y =11

⎧⎨⎩

Tue, Apr 12

EXAMPLE 2Solve the system of equations using Cramer’s rule (matrices).

3x − 7 y = −6

x + 2 y =11

⎧⎨⎩

A =3 −7

1 2

Tue, Apr 12

EXAMPLE 2Solve the system of equations using Cramer’s rule (matrices).

3x − 7 y = −6

x + 2 y =11

⎧⎨⎩

A =3 −7

1 2 Ax =

−6 −7

11 2

Tue, Apr 12

EXAMPLE 2Solve the system of equations using Cramer’s rule (matrices).

3x − 7 y = −6

x + 2 y =11

⎧⎨⎩

A =3 −7

1 2 Ax =

−6 −7

11 2 Ay =

3 −6

1 11

Tue, Apr 12

EXAMPLE 2Solve the system of equations using Cramer’s rule (matrices).

3x − 7 y = −6

x + 2 y =11

⎧⎨⎩

A =3 −7

1 2 Ax =

−6 −7

11 2 Ay =

3 −6

1 11

x =

Ax

A

Tue, Apr 12

EXAMPLE 2Solve the system of equations using Cramer’s rule (matrices).

3x − 7 y = −6

x + 2 y =11

⎧⎨⎩

A =3 −7

1 2 Ax =

−6 −7

11 2 Ay =

3 −6

1 11

x =

Ax

A =

(−6)(2) − (−7)(11)(3)(2) − (−7)(1)

Tue, Apr 12

EXAMPLE 2Solve the system of equations using Cramer’s rule (matrices).

3x − 7 y = −6

x + 2 y =11

⎧⎨⎩

A =3 −7

1 2 Ax =

−6 −7

11 2 Ay =

3 −6

1 11

x =

Ax

A =

(−6)(2) − (−7)(11)(3)(2) − (−7)(1)

=−12 + 77

6 + 7

Tue, Apr 12

EXAMPLE 2Solve the system of equations using Cramer’s rule (matrices).

3x − 7 y = −6

x + 2 y =11

⎧⎨⎩

A =3 −7

1 2 Ax =

−6 −7

11 2 Ay =

3 −6

1 11

x =

Ax

A =

(−6)(2) − (−7)(11)(3)(2) − (−7)(1)

=−12 + 77

6 + 7 =

6513

Tue, Apr 12

EXAMPLE 2Solve the system of equations using Cramer’s rule (matrices).

3x − 7 y = −6

x + 2 y =11

⎧⎨⎩

A =3 −7

1 2 Ax =

−6 −7

11 2 Ay =

3 −6

1 11

x =

Ax

A =

(−6)(2) − (−7)(11)(3)(2) − (−7)(1)

=−12 + 77

6 + 7 =

6513 = 5

Tue, Apr 12

EXAMPLE 2Solve the system of equations using Cramer’s rule (matrices).

3x − 7 y = −6

x + 2 y =11

⎧⎨⎩

A =3 −7

1 2 Ax =

−6 −7

11 2 Ay =

3 −6

1 11

x =

Ax

A =

(−6)(2) − (−7)(11)(3)(2) − (−7)(1)

=−12 + 77

6 + 7 =

6513 = 5

y =

Ay

A

Tue, Apr 12

EXAMPLE 2Solve the system of equations using Cramer’s rule (matrices).

3x − 7 y = −6

x + 2 y =11

⎧⎨⎩

A =3 −7

1 2 Ax =

−6 −7

11 2 Ay =

3 −6

1 11

x =

Ax

A =

(−6)(2) − (−7)(11)(3)(2) − (−7)(1)

=−12 + 77

6 + 7 =

6513 = 5

y =

Ay

A =

(3)(11) − (−6)(1)(3)(2) − (−7)(1)

Tue, Apr 12

EXAMPLE 2Solve the system of equations using Cramer’s rule (matrices).

3x − 7 y = −6

x + 2 y =11

⎧⎨⎩

A =3 −7

1 2 Ax =

−6 −7

11 2 Ay =

3 −6

1 11

x =

Ax

A =

(−6)(2) − (−7)(11)(3)(2) − (−7)(1)

=−12 + 77

6 + 7 =

6513 = 5

y =

Ay

A =

(3)(11) − (−6)(1)(3)(2) − (−7)(1)

=33 + 66 + 7

Tue, Apr 12

EXAMPLE 2Solve the system of equations using Cramer’s rule (matrices).

3x − 7 y = −6

x + 2 y =11

⎧⎨⎩

A =3 −7

1 2 Ax =

−6 −7

11 2 Ay =

3 −6

1 11

x =

Ax

A =

(−6)(2) − (−7)(11)(3)(2) − (−7)(1)

=−12 + 77

6 + 7 =

6513 = 5

y =

Ay

A =

(3)(11) − (−6)(1)(3)(2) − (−7)(1)

=33 + 66 + 7

=3913

Tue, Apr 12

EXAMPLE 2Solve the system of equations using Cramer’s rule (matrices).

3x − 7 y = −6

x + 2 y =11

⎧⎨⎩

A =3 −7

1 2 Ax =

−6 −7

11 2 Ay =

3 −6

1 11

x =

Ax

A =

(−6)(2) − (−7)(11)(3)(2) − (−7)(1)

=−12 + 77

6 + 7 =

6513 = 5

y =

Ay

A =

(3)(11) − (−6)(1)(3)(2) − (−7)(1)

=33 + 66 + 7

=3913 = 3

Tue, Apr 12

EXAMPLE 2

3x − 7 y = −6

x + 2 y =11

⎧⎨⎩

x = 5, y = 3

Tue, Apr 12

EXAMPLE 2

3x − 7 y = −6

x + 2 y =11

⎧⎨⎩

x = 5, y = 3

Check:

Tue, Apr 12

EXAMPLE 2

3x − 7 y = −6

x + 2 y =11

⎧⎨⎩

x = 5, y = 3

Check: 3(5) − 7(3) = −6

Tue, Apr 12

EXAMPLE 2

3x − 7 y = −6

x + 2 y =11

⎧⎨⎩

x = 5, y = 3

Check: 3(5) − 7(3) = −6

15 − 21 = −6

Tue, Apr 12

EXAMPLE 2

3x − 7 y = −6

x + 2 y =11

⎧⎨⎩

x = 5, y = 3

Check: 3(5) − 7(3) = −6

15 − 21 = −6 5 + 2(3) =11

Tue, Apr 12

EXAMPLE 2

3x − 7 y = −6

x + 2 y =11

⎧⎨⎩

x = 5, y = 3

Check: 3(5) − 7(3) = −6

15 − 21 = −6 5 + 2(3) =11

5 + 6 =11

Tue, Apr 12

EXAMPLE 2

3x − 7 y = −6

x + 2 y =11

⎧⎨⎩

x = 5, y = 3

Check: 3(5) − 7(3) = −6

15 − 21 = −6 5 + 2(3) =11

5 + 6 =11

(5,3)

Tue, Apr 12

PROBLEM SET

Tue, Apr 12

PROBLEM SET

p. 356 #1-31 odd

Solve all using matrices by hand

“I’m a great believer in luck, and I find the harder I work the more I have of it.” - Thomas Jefferson

Tue, Apr 12