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Question 1 4 '24 Suppose that a matrix A satisfies A 2 = 12 point *4 24 Enter an eigenvalue for A Number Question 2 Consider the following matrix A = (8 ,3) 1 point List the eigenvalues in increasing order, separated by a comma, including any that are repeated. For example, -1,-1, for a repeated eigenvalue or -1,1 for distinct onesQuestion 3 The matrix [ 01 1:] has 11 as one of its eigenvalues. Enter a corresponding eigenvector in the format (2,5) for example. 1 point You do not need to try and enter all solutions with a parameter, just choose one single eigenvector. @1 Question 4 Suppose that a 5 X 5 matrix is knnwn to have exactly two distinct eigenvalues, 9 and 2' If the trace of the matrix is 17 and the determinant is 144 then what is the multiplicity of the eigenvalue 9? 1 point Number Question 5 Suppose a matrix A has characteristic polynomial IAI - A| = (1 -9)2(1+9) 1 point What is the trace of A? Number21 - Question 6 Suppose that a matrix A has an eigenvalue -3 and corresponding eigenvector -45 1 point Which of the following is A -15 ? o [-Is] O -21 45 O 21 -45 O -63 135Question 7 For each of the following statements, select True or False (i) Every square matrix is diagonalisable 1 point O True 0 False (ii) If an n X n matrix is diagonalisable then it must have n distinct eigenvalues (i.e. all of them are different) 0 True O False (iii) If a matrix A is diagonalisable with P'IAP : D then the diagonal matrix D has the eigenvaues of A as the diagonal entries. 0 True 0 False Question 8 1 point 1 0 0 Suppose that A is diagonalised by P with P'IAP = 0 71 0 Which of the following would calculate A16? 1 0 0 OP 0 1 o P'1 0 0 65536 16 0 0 OP'1 0 716 0 P o 0 32 1 0 0 OP'1 0 1 0 P 0 0 65536 16 0 0 0P0 7160 P'1 0 032 0 0 2 Question 9 Suppose that a dynamic predator prey model predicts the number of feral cats and mice overtime by an equation Axk = xk+1 1 point ck: where xk : mk gives the number of cats and mice at time 1:. Suppose that A has eigenvalues A1 : 1.15 and A2 : 0.83 with corresponding eigenvectors v1 : [530] and v2 : [115%] By expressing x0 : c1 v1 + c2 v2 we have xk : c1 Alfvl + c2 Agvz and can determine the long term behaviour of the system. Which of the following is true in the long term? O Both cat and mice populations survive with a ratio of 150 mice per 10 cats 0 Both cat and mice populations die out 0 Only the cat population survives 0 Both cat and mice populations survive with a ratio of 500 mice per cat Question 10 2 2 Given a matrix 2 71 , use the PCA algorithm to find the covariance matrix C 1 point '3 '1 (Note that you must use the algorithm as given in the course notes, some other sources may give a different algorithm) . If any matrix entries are not whole numbers you should use fractions rather than decimals. To enter a matrix click on the 3X3 grid of squares belowt Next select the exact size of the matrix you want, Then change the entries in the matrix to the entries of your answer, It you need to start over then click on the trash can. First find the matrix XI X' _ b . 6 a 311101) f 00 a [2 Mi 0 (9m 1 then use that to find C C = b . 8 (1 sm (:1) 3 f 00 a [2 M o m