1 2 -1 0 0 1 0 0 -1 3 ſi 2 0 2 5 [10 (11 points) The matrix A= 2 1 3 2 7 reduces to R= 0 3 1 a 6 5 0 1 Let ui, , 13, 144, and us be the columns of U. (a) Determine, with justification, whether each of the following sets is linearly independent or linearly dependent. i. {u1, 12, 13) ii. {u1, 13, us} iii. {u2, 13} iv. {u1, 12, 13,...
ſi 0 1 37 14 00 11 1. Compute the determinant of 10 4 11 5 using cofactors. Show your work. 12 0 1 2
4. Find the eigenvalues of Aº for ſi 3 7 11] To 1 38 A= lo 7 0 4 LO 0 0 2
4. (3 points) Let ſi 2 1] A= 0 4 3 [1 2 2 Compute the third column of A-1 by solving the equation Ax = es, where ez = 0 Hint: Use Cramar's rule to solve the equation, noticing that the third column of A-' is given by the solution of the above equation. In fact there is nothing special about A-1, the third column of any 3 x 3 matrix B is given by the product Bez. Can...
Problem 5: Evaluate log(x) Jo 4+2 0 3. Show that 2x cos(e) Jo 1-cos(0) Problem 5: Evaluate log(x) Jo 4+2 0 3. Show that 2x cos(e) Jo 1-cos(0)
Question 5 (20 marks) a) [10 marks] Evaluate: 1 el-X Vx+y(y – 2x)2 dydx. 0 Jo (Hint: consider a change of variable.) b) [10 marks) Find the volume of the solid bounded by the sphere x2 + y2 + z2 = aand the cylinder x2 + y2 = ax, a > 0.
(2 points) Let -2 3 -1 -6 9 -3 Describe all solutions of Ax = 0. x = x2 +X4 +x3
(33 pts) This question is about the matrix = ſi 2 [3 2 0 4 1 6 3 1] 4 9 co (a) Find a lower triangular L and an upper triangular U so that A = LU. (b) Find the reduced row echelon form R = rref(A). How many independent columns in A? (c) If the vector b is the sum of the four columns of A, write down the complete solution to Ax = b
3.8 Find the general solution of ſi 2-3 47 10 -1 2 2 X= 0 0 0 1 How many parameters do you have?
0 ſi 1 19. (5 points) Find the eigenvalues and eigenvectors of A= 0 2 2 Lo 03 1 0 20. (5 points) Show that A= 0 2 2 is diagonalizable by finding P and D such that p-1AP = D for [003] a diagonal D.