2 1 4 6

5 3 0 9

1 8 1

3 3 5 2

6 0 4

Name: _________________________

1.

2.

3.6-7 Wups

3. 2 3 1 4 10 22

5 3 5 16 14

x

y

Corrected By: _________________________

3.6 Multiply Matrices3.7 Evaluate Determinants

By the end you should:1.describe matrix products2.multiply matrices3.solve matrix operations4.evaluate determinants5.find the area of triangle regions

Describing Matrix Products (multiplication)

You may only multiply two matrices together if the number of columns (vertical) in A equals the number of rows (horizontal) in B

45 8

7

Your Turn:

State if the product of AB is defined. If yes, then give the dimensions.

1. A: 4 x 3; B: 3 x 2 2. A: 5 x 2; B: 2 x 2

3. A: 3 x 2; B: 3 x 2 4. A: 3 x 5; B: 4 x 3

Multiply Matricesrow times column

add values together

Algebra:

Pictorial:

2 3 1 4

1 5 3 2

Numerical:

a b e f ae bg af bh

c d g h ce dg cf dh

Choo…

choo!

….to multiply matrices using your calculator

2 3 1 4

1 5 3 2

On your calculator…..Enter the matrices one in matrix A and one in matrix BFrom the home screen select matrix A, select matrix BPress ENTER

X

2. Using the same matrices from above. Find BA.

1. Find AB if

Your Turn:

1 4 5 7A= B=

3 2 9 6

Do AB and BA equal each other? _______

Matrix multiplication _____ ______ ____________________

• AssociativeA(BC) = (AB)C

• Left DistributiveA(B + C) = AB + AC

• Right Distributive(A + B)C = AC + BC

• Associative of a Scalark(AB) = (kA)B = A(kB)

Remember that

ORDER MATTERS

in matrix

multiplication

Properties of Matrix Multiplication

Matrix Operations

Using the given matrices, evaluate the following expressions.

3 -22 3 2 1

A= 0 4 B= C=1 0 4 2

-1 5

1. A(B + C)

2. A(B - C)

Two hockey teams submit equipment lists for the season as shown. Each stick costs $60, each puck costs $2, and each uniform costs $35. Use matrix multiplication to find the total cost of equipment for each team.

Real Life MatricesEQUIPMENT LISTS

Women's Team14 sticks30 pucks

18 uniforms

Men's Team16 sticks25 pucks

20 uniforms

inventory cost per item total cost

matrix matrix matrix

X

determinant = a number

- can only be found when using square matrices ex. 2 x 2, 3 x 3, 4 x 4...etc.

- denoted by "det A" or |A| - remember "disco fever" to solve

Determinants Think DISCO DANCE !!!!

Determinant of a Matrix

5 3ex.

2 1

2 x 2 deta b a b

ad cbc d c d

1. Multiply downdiagonal elements

2. Subtract up diagonal

elements (multiplied)

6 2

1 4

Your Turn: evaluate the determinant

Determinants Think DISCO DANCE !!!!

note: multiply down diagonal elements and add them;subtract the sum of the products of the up diagonals3 x 3

det

a b c a b c a b

d e f d e f d e aei bfg cdh gec hfa idb

g h i g h i g h

4 2 0

1 1 2

2 5 3

4 1 2

3 2 1

0 5 1

Your Turn: evaluate the determinant BY HAND

Determinants can be used to find the area of triangles whose vertices are coordinate points. This might be used by map makers or another aerial profession.

1 1

2 2

3 3

11

Area = 12

1

x y

x y

x y

Point one

Point two

Point three

Note: area is always positive!

Off the coast of California lies a triangular region of the Pacific Ocean where huge sea lions and seals live. The triangle is formed by imaginary lines connecting Bodega Bay, the Farallon Islands, and Ano Nuevo Island. Use the determinant to estimate the area of the region.

Area of a Triangle

Dice Time!

• 1. Create a 4x4 matrix and find the determinant.

• 2. Create a 2x2, a 2x2, and a 2x3

A Linear Algebra PhysiqueYoda has a physique that is literally built for linear algebra. In order to operate this Jedi master by a computer as opposed to the hand of a puppeteer, the character must be digitally created via a wireframe or tessellation as seen above. The picture above (the head) is a detail of a model that uses 53,756 vertices. Below is a model containing 33,862 vertices. Note the additional smoothness resulting from the additional vertices. Both models are available below. The graphics on this web page required two pieces of information -- the location of each vertex and thevertices that determine each face. Armed with vertexand face information, we can move Yoda using simplematrix multiplication. Let V be the 33,862 by 3 matrixassociated with the wireframe seen to the right.Note that row i of V contains the x, y and zCoordinates of the ith vertex in the model. Theimage can be rotated by t radians about the y-axisby multiplying V with Ry where

The necessary computation is much larger than those generally performed in linear algebra classes. Since V and Ry are 33,862 by 3 and 3 by 3 matrices, respectively, one rotation of the image requires 304,758 multiplications.

http://www.davidson.edu/math/chartier/Starwars/default.html

Homework:3.6 Page 199 (#3 – 9 odd, 15, 17, 23 – 31 odd, 37, 38)3.7 pg. 207 (#3, 6, 12, 18 , 24, 27, 40) ****Do #6 and 12 BY HAND

ALEKs Suggestions:Systems of Linear Equations: Matrices –Finding the determinant of a 2x2 matrix AND Finding the determinant of a 3x3 matrix AND Cramer's rule: Ptype 1 AND Cramer's rule: Ptype 2

• For a 2x2 system (2 equations & 2 variables)A is the coefficient matrix for the linear system:

ax + by = ecx + dy = f

If det A ≠ 0 then the one solution of this system is:

e b a ex = f d and y = c f

det A det A

You can do the same with a 3x3 system by replacing the constants for each column of the coefficients

Cramer’s Rule(ALEKs)

Cramer’s rule: Ptype IUse Cramer’s rule to find the solution to the

following system of linear equations:

ALEKS

pcalc045

Cramer’s rule: Ptype 2Use Cramer’s rule to find the value of y that

satisfies the system of linear equations:

ALEKS

alge022

Exit Card 3.6-7

Discuss with your table partner…

Did you know that Yoda was so complicated?

Do you anticipate that we can solve systems of linear equations using matrices?

What other real life applications might use matrices?