Explanation Given information: A small school has 100 students and three classrooms A, B and C. In the first period half the students in room A move to room B, one-fifth of the students in room B move to room C, and one-third of the students in room C move to room A. Concept used: The inverse of matrix A with 3 × 3 order can be obtained by elementary row operation on a new matrix to change the left side of a matrix in to identity matrix. And automatically the right matrix transformed in to inverse of matrix A . Here, the matrix A is [ a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ] . The new matrix can be written as, [ a 11 a 12 a 13 1 0 0 a 21 a 22 a 23 0 1 0 a 34 a 32 a 33 0 0 1 ] The solution of the matrix equation can be calculated as, A X = B X = A − 1 B Here, A is a square n × n matrix that is invertible, B is known matrix with n rows and X is a unknown matrix. Calculation: Consider x as A , y as B and z as C of the classrooms. The school has 100 students and three classrooms A, B and C. x + y + z = 100 ........ (1) In the first period half the students in room A move to room B and one-third of the students in room C move to room A. 1 2 x + 1 3 z = x Further simplify the above equation 1 2 x − 1 3 z = 0 ........ (2) Now, full room B, with that missing half of room A combining with the remaining 4 5 that left for room C. 1 2 x + 4 5 y = y Further simplify the above equation 1 5 y − 1 2 x = 0 ........ (3) The one-fifth of the students in room B, with 2 3 left from 1 s t period and now in A, 1 5 y + 2 3 z = z Further simplify the above equation 1 3 z − 1 5 y = 0 ........ (4) Add equation (3) and (4). to obtain the equation (5) as written as, 1 5 y − 1 2 x + 1 3 z − 1 5 y = 0 Further simplify the above equation − 1 2 x + 1 3 z = 0 ........ (5) Add equation (2) and (3). to obtain the equation (6) as written as, 1 2 x − 1 3 z + 1 5 y − 1 2 x = 0 Further simplify the above equation − 1 3 z + 1 5 y = 0 ........ (6) Consider B as column matrix a small school has 100 students and three classrooms A, B and C can be represent as, B = [ 100 0 0 ] The equation (1), (5), and (6) change in matrix equation can be written as, [ 1 1 1 − 1 2 0 1 3 0 1 5 − 1 3 ] [ x y z ] = [ 100 0 0 ] The above matrix equation can be written as, [ x y z ] = [ 1 1 1 − 1 2 0 1 3 0 1 5 − 1 3 ] − 1 [ 100 0 0 ] ........ (7) The given matrix can be written in the form of new matrix as, [ 1 1 1 1 0 0 − 1 2 0 1 3 0 1 0 0 1 5 − 1 3 0 0 1 ] Perform the operation R 3 → − 3 2 ⋅ R 3 and R 2 → R 2 − 5 6 ⋅ R 3 in the above matrix as, [ 1 1 1 1 0 0 0 1 2 5 6 1 2 1 0 0 0 1 3 10 3 5 − 3 2 ] [ 1 1 1 1 0 0 0 1 2 0 1 4

7th Edition

ISBN: 9781305115545

Author: James Stewart, Lothar Redlin, Saleem Watson

Publisher: Cengage Learning

See similar textbooks

Concept explainers

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.

Videos

Question

Chapter 6.1, Problem 72E

To determine

To find:

The total number of students in each room is the same for both periods.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 6.1, Problem 71E

Chapter 6.1, Problem 73E

Students have asked these similar questions

Safari File Edit View History Bookmarks Window Help Ο Ω OV O mA 0 mW ర Fri Apr 4 1 222 tv A F9 F10 DII 4 F6 F7 F8 7 29 8 00 W E R T Y U S D பட 9 O G H J K E F11 + 11 F12 O P } [

So confused. Step by step instructions please

In simplest terms, Sketch the graph of the parabola. Then, determine its equation. opens downward, vertex is (- 4, 7), passes through point (0, - 39)

Chapter 6 Solutions

College Algebra

Ch. 6.1 - If a system of linear equations has infinitely...Ch. 6.1 - Write the augmented matrix of the following system...Ch. 6.1 - Prob. 3E Ch. 6.1 - Prob. 4E Ch. 6.1 - Prob. 5E Ch. 6.1 - Prob. 6E Ch. 6.1 - Prob. 7E Ch. 6.1 - Prob. 8E Ch. 6.1 - Dimension of a Matrix State the dimension of the...Ch. 6.1 - Prob. 10E

Knowledge Booster

$Algebra$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.

The total number of students in each room is the same for both periods.

Solution Summary: The author explains how the inverse of matrix A with 3times 3 order can be obtained by elementary row operation on a new matrix.

Concept explainers

Videos

To find: The total number of students in each room is the same for both periods.

Want to see the full answer?

Chapter 6 Solutions

To find:

The total number of students in each room is the same for both periods.