• University of California, Merced

• BIO 180

Homework #11: Linear Models of Age Structured Populations Goal: Be able to carry out mathematical operations with matrices, represent population difference equations using matrices, and use R to perform matrix operations including determining the eigenvalues and eigenvectors for matrices, and interpret their meaning. • Follow the instructions provided, filling in your answers as you go. Make sure the text of your answers is noticeably different from black Times New Roman 12 pt font (that’s what this is), so that your answers are recognizable without having to hunt for them. • Note that for much of Part A of this homework, you can either type your answers on a computer, or you can work by hand on paper. If you prefer to work by hand, you can simply take a legible picture of your work and insert it into the appropriate spot in the homework. • For Part B of the homework: you’ll be working in R and will need to insert a copy of your R transcript into the appropriate spot in your homework. Remember to use a font like Courier New to preserve the alignment of the text. • Be sure to include copies of any graphs you are asked to make. Steps that have an R output that you’ll need to copy/paste* are denoted with the  symbol. • As you work, you are allowed to discuss the homework with your peers, but note that you must complete all of the steps independently and answer all of the questions in your own words. Part A – Matrices and Difference Equations Problem 1 (Modified from problem 2.1.1 on Page 51 of the textbook) Do the following matrix multiplications, showing your work (calculation setups) as you go. a. ( 3 −4 7 2 ) ( 1 3 ) Your work here: ( 3 −4 7 2 ) ( 1 3 ) = (3 ? 1) +(−4 ? 3) (7 ? 1) + (2 ? 3) = ( −9 13) b.( −3 1 −1 0 −4 3 2 4 1 ) ( 5 2 6 ) Your work here: ( −3 1 −1 0 −4 3 2 4 1 ) ( 5 2 6 ) = (−3 ? 5) +(1 ? 2)+ (−1 ? 6) ( 0 ? 5) +(−4 ? 2)+ (3 ? 6) (2 ? 5) +(4 ? 2)+ (1 ? 6) = −19 10 24 2 c.( 3 2 2 4 ) ( 2 −3 1 7 ) Your work here: ( 3 2 2 4 ) ( 2 −3 1 7 ) = (3 ? 2)+ (2 ? 1) (3 ? − 3)+ (2 ? 7) (2 ? 2)+ ( 4 ? 1) (2 ? − 3)+ (4 ? 7) = 8 5 8 22 d. Optional for extra credit: Check your answers to part a. – c. by using matrix operations in R > x X Error: object 'X' not found > x [,1] [,2] [1,] 3 -4 [2,] 7 2 > y y [,1] [1,] 1 [2,] 3 > x%*%y [,1] [1,] -9 [2,] 13 > a a [,1] [,2] [,3] [1,] -3 1 -1 [2,] 0 -4 3 [3,] 2 4 1 > b b [,1] [1,] 5 [2,] 2 [3,] 6 > a%*%b [,1] [1,] -19 [2,] 10 [3,] 24 > c c [,1] [,2]