1. Determine whether the matrices A =   37 18 18 24 13 12 −100 −50 −49   and B =   30 16 15 17 11 9 −79 −44 −40   are diagonalisable. If either matrix are diagonalisable, write it in the form P DP −1 , where D is diagonal. 2. Find the matrix that represents the linear transformation of the plane obtained by reflecting in the line y = x, then rotating anticlockwise through an angle of 45 degrees, and finally reflecting in the y axis. Give a simpler geometrical description of what this transformation does. 3. Find the eigenvalues and eigenvectors of the symmetric matrix S =   −1 2 0 2 2 1 0 1 −1  . Let M be the 3 × 3 matrix whose columns are the eigenvectors you have found. Evaluate MTM, with as little computation as possible. Give reasons for any computations you were able to omit. 4. Factorize the denominator of f(x) = 2x 5 + 15x 4 + 15x 3 + 2x 2 + 2 x 5 + 2x 4 + x 3 − x 2 − 2x − 1 completely, and use this to write f(x) as a partial fraction. 5. This question is designed to allow you to demonstrate your ability to apply the skills you have acquired in studying linear algebra to a real world problem. In a herd of wild cattle, the females can be classified as being either calves (up to 1 year old), yearlings (between 1 and 2 years old) and adults. Each year, for every 100 adult females, 75 female calves are born. Each year, about 60% of the calves survive to become yearlings, 80% of the yearlings survive to become adults and 95% of the adults survive. So, if at some point in time there were 100 female calves, 100 female yearlings First downloaded: 8/4/2015 at 16:20::17 and 100 female adults, then, after a year there will be 75 female calves, only 60 of the original calves will have survived to become yearlings, and 80 of the original yearlings will have survived to join the 95 surviving adults to give 175 female adults. Suppose that at the start of a study, there are c0 female calves, y0 female yearlings and a0 female adults, and write cn, yn, and an for the corresponding numbers n years after the start of the study. (a) Write down equations for c1, y1 and a1 in terms of c0, y0 and a0 and express these equations in matrix form, using a matrix A (b) Use this to find a matrix equation that would enable you to calculate cn, yn and an from c0, y0 and a0. (c) Find the eigenvalues of A and explain why the matrix is diagonalisable. (d) For each eigenvalue λ, calculate |λ|. (e) If   c0 y0 a0   = αv1 + βv2 + γv3, where v1, v2 and v3 are eigenvectors corresponding to eigenvalues λ1, λ2 and λ3, write down an expression for   cn yn an   in terms of α, β, γ and the eigenvectors and eigenvalues. (f) Interpret your results to give information about the long term composition of the herd and its rate of growth or decline. (g) What would happen to the herd if an enviromental factor dramatically reduced fertility so that only 10 female calves were born each year per 100 adult females? Explain your reasoning. (You can use technology to find the roots of the characteristic polynomial of the matrix you encounter if you wish). 6. This question applies the ideas of linear algebra to fitting a graphs to data. (a) Suppose I want to find a quadratic equation of the form y = a + bx + cx2 to pass through the points (−2, 25), (3, 0) and (6, 33). Explain how this is related to the matrix equation   1 −2 4 1 3 9 1 6 36     a b c   =   25 0 33   and hence use matrix techniques to find a, b and c. Interpret your solution in terms of writing   25 0 33   as a linear combination of the vectors that form the columns of the matrix. (b) Write down the matrix equation A   a b c   = v to solve if I also want the quadratic equation to go through (5, 10). Express the fact that this is not possible in terms of a vector not being a linear combination of the vectors that form the columns of a matrix. (c) Explain why, if v = u+w where u is a linear combination of the columns of A and w is orthogonal to each of the columns of A, then AT v = ATu, and hence deduce that if   a0 b0 c0   is a solution of AT A   a b c   = AT v, then A   a0 b0 c0   = u and the error w is as small as possible. (d) Use technology to solve the appropriate matrix equation to find the quadratic equation that is the best fit to the points (−2, 25), (3, 0), (5, 10) and (6, 33). Discuss whether this is a better fit to the data than the quadratic you obtained in part (a). (e) Often data is expected to follow an exponential growth model of the form y = Aekt, where t measures time and k is called the growth rate. By rewriting the equation as log y = kt + log A, use this technique to find the values of A and k that give the best fit of the exponential growth model to experimental data where the values of y at times 0, 1, 2 and 3 are 11, 23, 42 and 80 respectively