MATH 33 Sample Questions for Exam 3. Find x and y so that x 4 3 5x 3y + y = 5 5. x = 3/7, y = 49/7. Let A = 3 4, B = 3 5, C = 3 Perform the indicated operations, if possible: a AC b AB c B + AC d CBA AB is not defined, the others are defined. 3 3. Show that the matrix A = 5 3 is nonsingular and find the element in the,3 8 position of A. det A =. Therefore A is nonsingular. If AB = I, then b 3 = 3. 4. Show that the matrix of coefficients is non-singular and use Cramer s rule to find the unique solution of the system:. x x + x 3 = x x 3 = x +3x = x =5,y= 3, z = 7 5. Write the system in the vector-matrix form Ax = b and solve by finding A. x =5,x = 8, A = x + x = 4x +3x = 4 3/ / 6. Write the system in the vector-matrix form Ax = b and solve by finding A. x + x = x +3x x 3 = x +x 3 =4
x =8,x = 6, x 3 = A = 7. Given the set of vectors 6 5 3 {u =,,, v =,,, w =7, 4, } Is the set linearly dependent or linearly independent? If it is linearly dependent, express one of the vectors as a linear combination of the other two. Dependent, w =3u +v 8. Given the set of vectors {v =,,, v = 3,,, v 3 =8,, 5, v 4 = 9,, 5} Is the set linearly dependent or linearly independent? If it is linearly dependent, how many independent vectors are there in the set? Dependent, 9. For what values of a are the vectors v =a,,, v =, a, 3, v 3 =, a,, v 4 =3a,, a linearly dependent? All real numbers.. For what values of a are the vectors linearly dependent? u =a,,, v =, a, 3, w =, a, a =4,. Given the set of functions {f x =+x, f x = x, f 3 x =x }. Calculate the Wronskian of the functions. Is the set linearly dependent or linearly independent? +x x x x = 4 ; independent. Given the set of functions {f x=+x, f x = x, f 3 x =3x }. Calculate the Wronskian of the functions. Is the set linearly dependent or linearly independent? If it is linearly dependent, express one of the functions as a linear combination of the other two. +x x 3x 3 = ; I believe the set is dependent f 3 = 3 f 5 3 f.
3. The eigenvalues and eigenvectors of 4, ;, 4. The eigenvalues and eigenvectors of, ;,, 5. The eigenvalues and eigenvectors of 3, ;, 6. The eigenvalues and eigenvectors of +3i, 5 + i 3 6 4 4 ;, 3 3 3 4 ;, 5 3 3i, are: are: are: are: 5 i Hint: 4 is an eigenvalue. Hint: is an eigenvalue. Hint: is an eigenvalue. 7. The general solution of y 4 y +y +6y = is: Hint: r = is a root of the characteristic equation 3 y = C e x + C e 3x + C 3 e x 8. The general solution of y +y 8 y y = is: Hint: r = 3 is a root of the characteristic equation y = C e 3x + C e x + C 3 xe x 9. The general solution of y 4 +y +4y y 5 y = is: Hint: r = +i is a root of the characteristic equation y = C e x cos x + C e x sin x + C 3 e x + C 4 e x. The homogeneous equation with constant coefficients of least order that has y =e 3x + 3 sin x +x as a solution is: 3
y 5 3y 4 +4y y =. A particular solution of y 4 6 y =e x +3e 4x + cos x + 5 will have the form: z = Axe x + Be 4x + Cxcos x + Dx sin x + E. A particular solution of y 3y y =4e 5x +e x + 6 will have the form: z = Axe 5x + Be x + Cx 3. Given the differential equation y 4y 3y +8y =. a Write the equation in the vector-matrix form x = Ax. x = x 8 3 4 b Find three linear independent solution vectors of the system in a, given that is a root of the characteristic polynomial. v = e t, v = e 3t 3, v 3 = e 3t + te 3t 3 4 9 6 9 4. Find the solution of the initial-value problem x = is a root of the characteristic polynomial. General solution: x = C e 3t + C e t 3 Solution of the initial-value problem: x =e t 5. Find a fundamental set of solutions of x = {e t [ cos t 6. Find the general solution of x = polynomial. sin t 4 x. ],e [cos t t x; x = + C 3 e t e t + sin t ;. HINT: ]} x. HINT: is a root of the characteristic 4
x = C e t 3 4 + C e t + C 3 e t + te t 7. Find a fundamental set of solutions of x = characteristic polynomial. et,e t,e t 5 4 4 5 x. HINT: is a root of the 8. If the matrix of coefficients of a system of n linear equations in n unknowns does not have an inverse, then the system has no solutions. a Always true b Sometimes true. c Never true. d None of the above. b 9. If the rank of the matrix of coefficients of a homogeneous system of n linear equations in n unknowns is n, then the system has infinitely many solutions. a Always true b Sometimes true. c Never true. d None of the above. a 3. If the rank of the augmented matrix of a system of n linear equations in n unknowns is greater than the rank of the matrix of coefficients, then is not an eigenvalue of the matrix of coefficients. a Always true b Sometimes true. c Never true. d None of the above. c 5
3. If is not an eigenvalue of the matrix of coefficients of a homogeneous system of n linear equations in n unknowns, then the system does not have infinitely many solutions. a Always true b Sometimes true. c Never true. d None of the above. a 3. If a system of n linear equations in n unknowns is inconsistent, then the reduced row echelon form of the matrix of coefficients is not I n. a Always true b Sometimes true. c Never true. d None of the above. a 33. If a homogeneous system of n linear equations in n unknowns has no nontrivial solutions, then the determinant of the matrix of coefficients is positive. a Always true b Sometimes true. c Never true. d None of the above. b 6