The following statements are needed to prove Theorem 6.12. a. Show that if A~l exists

Perform the following matrix-vector multiplications: 2 1 3 b. 2 -2 " 1 -4 3 -2 _ -4 4 1

Perform the following matrix-vector multiplications: I a. c. 3 2 2 1 0 0 1 1 -1 2 -2 b. "3 2 ' 1 1 -1 0 2 4 ' 2 ' "3-20 5 d. [2-21] -2 3

Perform the following matrix-matrix multiplications: I 5 a. c. 2 3 2 4 5 -3 -I -3 3 2 2 0 b. 2 3 -3 -I 1 -3 5 2 -4 0 1 1 (N 0 1 "2 12" " 1 -2 0 1 0 -1 d. -2 3 0 -4 1 -4 2 3 -2 2 -1 3 0 2

Perform the following matrix-matrix multiplications: a. c. -2 3 " 2 -5 " " -1 3 " 2 -2 3 " b. 0 3 _ -5 2 -2 4 3 2 2 _ 2 -3 -2 " 2 -3 4 3 -1 0 " " -1 2 -3 4 1 -3 4 -1 d. 2 -2 3 4 -1 -2 1 -4 4 -1 -2 -2 1 4 3 _

Determine which ofthe following matrices are nonsingular and compute the inverse ofthose matrices: a. 4 3 2 0 6 7 -2 -1 -3 b. 1 2 3 2 0 1 -1 I I 1 1 -1 1 ' 4 0 0 0 c. 1 2 -4 -2 d. 6 7 0 0 2 1 1 5 9 11 1 0 -1 0 -2 -4 5 4 1 1

Determine which ofthe following matrices are nonsingular and compute the inverse ofthose matrices; a. c. 1 2 -1 " 4 1 0 0 0 1 2 b. 0 0 0 -1 4 3 _ 0 0 3 1 2 3 4 " " 2 0 1 2 2 1 -1 1 A 1 1 0 2 -3 2 0 1 a. 2 -1 3 1 0 5 2 6 3 -1 4 3

Given the two 4x4 linear systems having the same coefficient matrix: X\ X2 + 2x3 X4 6, x1 - X3+ X4 - 4, 2X| + X2 + 3x34x4 = 2, X2 + X3 X4 = 5; X| X2 + 2X3 X4 1, X| - X3+ X4 = 1, 2xi + X2 + 3X34X4 = 2, X2 + X3 X4 1.a. Solve the linear systems by applying Gaussian elimination to the augmented matrix 8. 9. 1 -1 2 -1 6 1 1 0 -1 1 4 1 2 1 3 -4 -2 2 0 -1 1 -1 5 -1 b. Solve the linear systems by finding and multiplying by the inverse of A = c. Which method requires more operations?

Consider the four 3x3 linear systems having the same coefficient matrix; 1 -1 2 -1 1 0 -1 1 2 1 3 -4 0 -1 1 -1 2x\ 3x2 + *3 2, X\+ X2- *3 = -1, x\ + X2 3x3 = 0; 2x| 3x2 + X3 = 0, X| + X2 - X3 = 1, X| + X2 3x3 - 3; 2xi 3x2 + -*3 = 6. X| + X2 - X3 = 4, X] + X2 3x3 = 5; 2xi 3x2 + X3 = 1, l + *2 X3 = 0, X| + X2 3X3 0. a. Solve the linear systems by applying Gaussian elimination to the augmented matrix 2 -3 1 | 2 6 0 -1 1 1 -1 : -1 4 1 0 1 1 -3 : 0 5 -3 0 b. Solve the linear systems by finding and multiplying by the inverse of A = 2 -3 I 1 1 -1 -1 I -3 c. Which method requires more operations?

It is often useful to partition matrices into a collection ofsubmatrices. For example, the matrices " 1 2 -1 ' 2-1 7 0 A = 3 -4 -3 and B = 3 0 4 5 6 5 0 -2 1-31 can be partitioned into 1 2 -1 3 -4 -3 6 5 0 -^ii A21 An A22 2 -1 7 0 and 3 0 4 5 -2 1 -3 1 fin : fii2 fi21 : B22 a. Show that the product of A and B in this case is AB = AUBU + A12S21 ^21 Si! + 422fi2l An fi|2 + Ai2fi22 A21 fil2 + A22B22 b. If B were instead partitioned into 2 -1 7 ; 0 B = 3 0 4 : 5 -2 1 -3 : 1 fin fi2. fi|2 B22 would the result in part (a) hold? c. Make a conjecture concerning the conditions necessary for the result in part (a) to hold in the general case

The study of food chains is an important topic in the determination of the spread and accumulation of environmental pollutants in living matter. Suppose that a food chain has three links. The first link consists ofvegetation oftypes U], V2, , u, which provide allthe food requirementsfor herbivoresof species 1,112,... ,/r, in the second link. The third linkconsistsofcarnivorous animals ci, C2,... .c*, which depend entirely on the herbivores in the second link for their food supply. The coordinate a,; of the matrix -4 = a\\ a 2\ a\2 22 Aim L flnl an2 represents the total number of plants oftype u, eaten by the herbivores in the species h whereas h. in B = fin fi2| J tn\ fi| 2 fi22 b\k bik bml hmk . describes the number of herbivores in species fi, that are devoured by the animals oftype Cj. a. Show that the number of plants oftype u, that eventually end up in the animals ofspecies cj is given by the entry in the ith row and Jth column ofthe matrix AB. b. What physical significance is associated with the matrices A -1 , B~[ , and {AB)~] B~l A~le!

In a papertitled "Population Waves," Bernadelli [Ber] (see also [Se]) hypothesizes a type ofsimplified beetle that has a natural life span of 3 years. The female of this species has a survival rate of ^ in the first year of life, has a survival rate of ^ from the second to third years, and gives birth to an average of six new females before expiring at the end of the third year. A matrix can be used to show the contribution an individual female beetle makes, in a probabilistic sense, to the female population of the species by letting a,-7- in the matrix A [a/,-] denote the contribution that a single female beetle of age j will make to the next year's female population of age i; that is, A = 0 0 6 1 0 0 Oil) a. b. c. The contribution that a female beetle makes to the population 2 years hence is determined from the entries of A 2 , of 3 years hence from A 3 , and so on. Construct A 2 and A 3 and try to make a general statement about the contribution of a female beetle to the population in n years' time for any positive integral value ofn. Use your conclusions from part (a) to describe what will occur in future years to a population of these beetles that initially consists of 6000 female beetles in each ofthe three age-groups. Construct A -1 and describe its significance regarding the population ofthis species.

Prove the following statements or provide counterexamples to show they are not true. a. The product of two symmetric matrices is symmetric. b. The inverse of a nonsingular symmetric matrix is a nonsingular symmetric matrix. c. If A and B are n x n matrices, then (AB)' = A'B'

The following statements are needed to prove Theorem 6.12. a. Show that if A~l exists, it is unique. b. Show thatif A is nonsingular, then (A-1) -1 = A. c. Show that if A and B are nonsingular n x n matrices, then (AS)-1 = S _1A _I

a. Show thatthe product oftwo n x n lower-triangular matrices is lower triangular. b. Show that the product oftwo n x n upper-triangular matrices is upper triangular. c. Show that the inverse of a nonsingular n x n lower-triangular matrix is lower triangular

In Section 3.6, we found that the parametric form (x(t), y(t)) of the cubic Hermite polynomials through (x(0), y(0)) = (xq, T'o) and 00). yO)) = Oi, )'i) with guidepoints (xq + yo + A)) and (xi ], yi A), respectively, are given by x(0 = (2(xo xO + (ao + ai))/3 + (3(xi - xo) - a, - 2ao)f2 + ao' +-*0 and y(0 = (2(yo - yO + (A) + /if)) t 3 + (3(yi - yo) - A - 2Ai) ' 2 + A' + yoThe Bezier cubic polynomials have the form x(0 = (2(x0 - xO + 3(a0 -I- a,)) 1 3 + (3(xl - xq) - 3(ai + 2ao)) r + 30-0' + -^o and y(0 = (2(yo - yi) + 3(A + fa)) ' 3 + (3(yi - yo) - 3(A + 2A)) t 2 + 3fat + yoa. Show that the matrix 7 4 4 0 -6 -3 -6 0 0 0 3 0 0 0 0 1 transforms the Hermite polynomial coefficients into the Bezier polynomial coefficients, b. Determine a matrix B that transforms the Bezier polynomial coefficients into the Hermite polynomial coefficients.

Suppose m linear systems Ax*'1 ' b 1 ' 1 ', P 1,2,... , m, are to be solved, each with the n x n coefficient matrix A. a. Show that Gaussian elimination with backward substitution applied to the augmented matrix [ A : b (1)b<2) b"'" ] requires I , 7 1 -n + inn -n multiplications/divisions and 1 , 7 1 9 1 -n + mn n mn Hn additions/subtraction b. Show that the Gauss-Jordan method (see Exercise 14, Section 6.1) applied to the augmented matrix A : b (1)b <2) b"'" ] requires and 1 , ,1 -zr + mn n multiplications/divisions 2 2 I , , /1 -n + (m \)n +(-?) additions/subtractions. c. For the special case b"" = 0 1 pth row, for each p = ,m, with m = n, the solution x 1 ''' is the pth column of A Show that Gaussian elimination with backward substitution requires 4 1 -n3 -n multiplications/divisions and 4 , 3 , 1 -n n Hn additions/subtractions 3 2 6 for this application and that the Gauss-Jordan method requires 3 , I -n n multiplications/divisions 2 2 F and 3 , 2 1 -n 2n Hn additions/subtractions. 2 2 d. Construct an algorithm using Gaussian elimination to find A -1 but do not perform multiplications when one of the multipliers is known to be 1 and do not perform additions/subtractions when one ofthe elements involved is known to be 0. Show that the required computations are reduced to n 3 multiplications/divisions and n 3 2n2 + n additions/subtractions. e. Show that solving the linear system Ax b, when A -1 is known still requires n 2 multiplications/divisions and n 2 n additions/subtractions. f. Show that solving m linear systems Ax1 ''' = b (p) , for /> = 1,2,... , m, by the method x (/' ) = A-'b'''1 requires mn2 multiplications and m(n2 n) additions if A -1 is known. g. Let A be an n x n matrix. Compare the number ofoperations required to solve n linear systems involving A by Gaussian elimination with backward substitution and by first inverting A and then multiplying Ax = bby A -1 , forn = 3, 10, 50, and 100. Is it ever advantageous to compute A -1 for the purpose of solving linear systems?

Use the algorithm developed in Exercise 16(d) to find the inverses of the nonsingular matrices in Exercise 5

Considerthe 2x2 linearsystem (A + i B)(x +iy) = c+/d with complex entries in component form: (flu + ibu )ix\ + jyi) + (012 + ibi2)(x2 + />'2) = C] + id\, (2i + ib2i){X] + iyi) + ((122 + ib22)(x2 + iyi) = ^2 + idia. Use the properties of complex numbers to convert this system to the equivalent 4x4 real linear system A\ By = c, Bx + Ay = d. b. Solve the linear system (1 2i)(x\ + iyi) + (3 4- 2/)(x2 + iya) = 5 + 2i, (2 + i)(x i + iyi) + (4 + 3i)(x2 + iyi) = 4- i.