Name: _________________________ Wups 3. Corrected By: _________________________

3.6 Multiply Matrices 3.7 Evaluate Determinants By the end you should: 1.describe matrix products 2.multiply matrices 3.solve matrix operations 4.evaluate determinants 5.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

Your Turn: State if the product of AB is defined. If yes, then give the dimensions. 1. A: 4 x 3; B: 3 x 22. A: 5 x 2; B: 2 x 2 3. A: 3 x 2; B: 3 x 24. A: 3 x 5; B: 4 x 3

Multiply Matrices row times column add values together Algebra: Pictorial: Numerical: Choo…choo!

….to multiply matrices using your calculator On your calculator….. Enter the matrices one in matrix A and one in matrix B From the home screen select matrix A, select matrix B Press ENTER X

2. Using the same matrices from above. Find BA. 1. Find AB if Your Turn: Do AB and BA equal each other? _______ Matrix multiplication _____ ______ ____________________

Associative A(BC) = (AB)C Left Distributive A(B + C) = AB + AC Right Distributive (A + B)C = AC + BC Associative of a Scalar k(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. 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 Matrices EQUIPMENT LISTS Women's Team 14 sticks 30 pucks 18 uniforms Men's Team 16 sticks 25 pucks 20 uniforms 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

2 x 2 1.Multiply down diagonal elements 2. Subtract up diagonal elements (multiplied) 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 diagonals 3 x 3 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. 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 Physique Yoda 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 the vertices that determine each face. Armed with vertex and face information, we can move Yoda using simple matrix multiplication. Let V be the 33,862 by 3 matrix associated with the wireframe seen to the right. Note that row i of V contains the x, y and z Coordinates of the ith vertex in the model. The image can be rotated by t radians about the y-axis by 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.

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 = e cx + dy = f If det A ≠ 0 then the one solution of this system is: e b a e x =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 I Use Cramer’s rule to find the solution to the following system of linear equations: ALEKSALEKS pcalc045

Cramer’s rule: Ptype 2 Use Cramer’s rule to find the value of y that satisfies the system of linear equations: ALEKSALEKS alge022

Exit Card 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?