C HAPTER 4 Matrices

4.1 I NTRO TO M ATRICES Matrix: a rectangular array of variables or constants in horizontal rows and vertical columns, usually enclosed in brackets Element: a value in a matrix Dimensions: number of rows x number of columns Read “m by n”

T YPES OF M ATRICES

S OLVE AN EQUATION INVOLVING M ATRICES

3. Write a matrix for the prices of movie tickets for adults, children, and seniors. What are the dimensions of the matrix? NOW PLAYING Ticket Information Evening ShowsMatinee Shows Adult …. $7.50Adult …. $5.50 Child ….$4.50 Senior….$5.50 Twilight Shows All tickets…….$3.75

4.2 O PERATIONS WITH MATRICES

S CALAR M ULTIPLICATION

4.3 M ULTIPLYING MATRICES

You can multiply matrices if and only if: the number of columns in the first matrix is the same as the number of rows in the second matrix Ex: A 5 x 3 and B 3 x 4 = AB If the matrices cannot be multiplied = product matrix is not defined 5 x 4

F IND RS IF

A T A SWIMMING MEET 6 POINTS ARE AWARDED FOR 1 ST PLACE, 4 POINTS FOR 2 ND PLACE, AND 3 POINTS FOR 3 RD PLACE. 2. The chart shows how many swimmers placed in each position through the meet for the four participating schools. Write a set of matrices to model the points earned. Which team won the meet? School1 st Place2 nd Place3 rd Place Central Dauphin473 Cumberland Valley881 Hershey1053 Carlisle336

C OMMUTATIVE P ROPERTY – D OES IT WORK FOR MATRICES ?

D ISTRIBUTIVE P ROPERTY – D OES IT WORK FOR MATRICES ?

4.5 D ETERMINANTS

F IND THE VALUE OF THE DETERMINANT

F IND THE DETERMINANT USING EXPANSION BY MINORS

4.6 C RAMER ’ S RULE

S OLVE THE SYSTEM OF EQUATIONS USING C RAMER ’ S RULE 1. 5x + 4y = 28 3x – 2y = 8

2. 2x – 3y = 12 -6x + y = -20

I N VOTING FOR THE COLORS OF A NEW HIGH SCHOOL, BLUE & GOLD RECEIVED 440 VOTES FROM 10 TH AND 11 TH GRADERS WHILE RED & BLACK RECEIVED 210 VOTES FROM THE SAME GRADES. I N THE 10 TH GRADE, BLUE & GOLD RECEIVED 72% OF THE TOTAL AND R ED & BLACK RECEIVED 28%. I N THE 11 TH GRADE, B LUE & GOLD RECEIVED 64% OF THE TOTAL AND R ED & B LACK RECEIVED 36%. Write a system of equations that represents the total number of votes for each pair of colors. Find the total number of votes cast in 10 th grade and in 11 th grade.

4.7 I DENTITY AND I NVERSE M ATRICES

Inverse Matrices: When the product of two matrices with the same dimensions is the identity matrix

D ETERMINE WHETHER EACH PAIR OF MATRICES ARE INVERSES OF EACH OTHER.

F IND THE INVERSE FOR THE GIVEN MATRIX

4.8 U SING M ATRICES TO SOLVE SYSTEMS OF EQUATIONS

S OLVE THE SYSTEM USING MATRICES 1. 5x + 3y = 13 4x + 7y = -8

2. 6a – 9b = -18 8a – 12b = 24