Ch+4+0910

Real Life Applications: Matthew Aguilar Notes 4-5: Katie Conner Example Problems: Cody Windsor Vocab: Austin Gibbons editor/ video: Cheyenne Bohlen notes 1-3:Haley Brandt A matrix is a rectangular arrangement of numbers in rows ad columns. For instance, matrix A below has two rows and three columns. The dimensions of this matrix are 2x3 (read 2 by 3). The numbers in a matrix are its entries. In matrix A, the entry in the second row, third column is 5. __ Properties of matrix operations __ Let A, B, and C be matrices of the same dimensions and let d be scalar.

Associative Property of Addition (A+B)+C=A+(B+C) Commutative Property of Addition A+B=B+A

Multiplication of a sum or difference of a matrix follows the distributive property.

Distributive Property of Addition d(A+B)=dA=dB Distributive Property of Subtraction d(A-B)=dA-dB

__ Multiplying Matrices __ The product of two matrices is only defined if the number of columns in A is equal to the number of rows in B. If A has a dimension of mxn and B had a dimension of nxp then the product of AB is mxp matrix. There is a specific pattern for multiplying matrices. The first number of the first row of the A matrix is multiplied by the first number of the first column of the B matrix. Next multiply the second number of the first row of the A matrix by the second number of the first column of the B matrix. Then add the products ae and bg. This is the first entry in your new matrix, C. ae + bg = i Next multiply the first number of the first row in matrix A by the first number in the second column of matrix B. Then multiply the second number of the first row in matrix A by the second number in the second row of matrix B. Add the products af and bh. This becomes the second number of the first row in your new C matrix. af + bh = j Take the first number of the second row of matrix A and multiply it by the first number in the first column of matrix B  Next take the second number of the second row of matrix A and multiply it by the second number of the first column of the B matrix. Next take the product ce and add it to dg. This is the third number in your new matrix. Take the first number of the second row of matrix A and multiply it by the first number of the second column of matrix B. Take the second number of the second row of matrix A and multiply it by the second number of the second column of matrix B. Now add the products cf and dh. This is the last number of your matrix. __ Multiplication Properties of matrices __

Assosiative Property of matrix multiplication A(BC)=(AB)C Left Distributive Property A(B+C)=AB=AC Rigtht Distibutive property (A+B)C=AC+BC Asssosiative Property of Scalar Multiplication c(AB)=(cA)B=a(cB) __ Determinant of a Matrix __ The determinant of a matrix is the difference of the products of the entries on the diagonals.

 Notes 4.4: Inverses and Identities.

The number one is the multiplicative identity for all real numbers because 1*b=b and b*1=b. The "n X n" Identity Matrix is that matrix that has 1's o the main diagonal and 0's everywhere else. ex. for a 2 X 2 matrix : If B is any matrix and I is the identity matrix, then IB=B and BI=B two n x n matrices are inverses of each other if their product (in both orders) is the n x n identity matrix. Ex. [3 -1][2 1] = [1 0] [-5 2] [5 3]= [0 1]
 * 1 || 0 ||
 * 0 || 1 ||
 * 0 || 1 ||

Notes 4.5: Solving Equations with Matrices. 1. Find the determinent of [A] 2. Muptiply [A] the reciprocal of the det. 3. In the new Matrix [A^(-1)], switch the a&d positions and the c&b signs. 4. Multiply [A^(-1)] by [B] to get [X]

[A]= [ 2 -1] [2 -1]

[B]= [1 2] [4 4]

Det of [A] =-4

[A^(-1)]=(-1/4)[2 -1]=[-0.5 0.25]= [0.25 -0.25] ....................[2 -1] =[-0.5 0.25]=[0.5 -0.5]

[A^(-1)][B]=[X]

[X]=[-0.75 -0.5] ......[-1.5 -1]

Real Life Application 4.1

Data of sales records, such as for CD’s or DVD’s, can be organized into a matrix to represent the data.

Real Life Application 4.2

Matrices can be set up to determine the cost of operation or the cost of equipment for a business. Because multiplying matrices requires only the multiplication of corresponding sectors of the matrix, cost and quantity can be multiplied together for cost.

Real Life Application 4.3

The area of a triangle can be determined with matrices, which can be used in architecture or landscaping. The matrix is set up where the first column of entries is the x-value of the triangle's coordinates, while the second column is the y-value. The third column is then filled with 1's. By finding the determinant, the area can be found after dividing by 2 and using the absolute value.

Example Problems 4.1

+ = Example Problems 4.2
 * 4 || 9 ||
 * 17 || 12 ||
 * 16 || 11 ||
 * 3 || 8 ||
 * 20 || 20 ||
 * 20 || 20 ||


 * Vocab 4.1 **


 * Matrix**- a rectangular arrangement of numbers in rows and columns. [ 3 4 ]
 * Dimensions**- the number //m// of rows of a matrix by the number //n// of columns of the matrix, written //m// x //n. [ 3 4 ]= 1 x 2//
 * Entries**- the numbers in a matrix. 3, 4
 * Equal**- when the dimensions are the same and the entries in corresponding positions are equal in the matrices.
 * Scalar**- a real number by which you multiply a matrix. 2[ 3 4 ]
 * Scalar Multiplication**- the process of multiplying each entry in a matrix by a scalar.


 * Vocab 4.3 **


 * Determinant**- a real number associated with any square matrix //A,// denoted by det //A// or by the absolute value of //A.//
 * Cramer's Rule**- a method for solving a system of linear equations which uses determinants of matrices.
 * Coefficient Matrix**- a matrix consisting of the coefficients of the variables in a set of linear equations.


 * Vocab 4.4 **


 * Identity Matrix**- the matrix that has 1's on the main diagonal and 0's elsewhere.
 * Inverses**- two //n// x //n// matrices are inverses of each other if their product (in both orders) is the //n// x //n// identity matrix.


 * Vocab 4.5 **
 * Matrix of Variables**- the matrix of variables of the linear system //ax// + //by// = //e//, //cx// + //dy// = //f// is [//x// over //y//][[image:http://www.wikispaces.com/i/c.gif width="22" height="22" caption="Italic"]].
 * Matrix of Constants**- the matrix of constants of the linear system //ax// + //by// = //e//, //cx// + //dy// = //f// is [e over f].