MATH 203, Matrix Algebra Review sheet for the final exam
(1) Reduce the following matrix A to reduced row-echelon form 0 0 −2 0 7 12 A = 2 4 −10 6 12 28 2 4 −5 6 −5 −1 (2) For the following matrix in reduced row-echelon form find the complete set of solutions of Ax = 0. 1 2 −3 0 0 5 0 0 0 0 1 0 −4 0 A= 0 0 0 0 1 6 0 0 0 0 0 0 0 1 Assuming that Ax = b has at least one solution, how many free variables can you expect for the general solution? (3) If A, B and C are all invertible n × n matrices, show that ABC is also invertible and find its inverse. Is it always true that if ABC is invertible, then all of A, B and C are invertible? (4) Show that for any n × n matrix A and a polynomial f (x), then f (A) is also an n × n matrix. Assume that B be similar to A, so B = P −1 AP . Show that for any polynomial f (x) we have f (B) = P −1 f (A)P . (5) Are the functions sin x, sin 2x and sin 3x linearly independent? (6) Use the fact that all the integers 1599, 1716, 2769 and 3003 are divisible by 13 to show that the following determinant is also divisible by 13, without directly evaluating the determinant 1 1 2 3 (Hint! Use Cramer’s Rule). 5 7 7 0 9 1 6 0 9 6 9 3
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(7) Use the fact that λ = 4 is an eigenvalue for the following matrix A to find all the eigenvalues for A if 0 1 0 A= 0 0 1 . 4 −17 8 (8) Diagonalize the following matrix 3 −2 0 3 0 . A = −2 0 0 5 (9) Is the following matrix diagonalizable? A= −3 2 −2 1 .
(10) Compute A2009 for the following matrix 1 3 3 A = −3 −5 −3 . 3 3 1 (11) Let (an )∞ be a number sequence given recursively by n=1 a1 = 1 a2 = 5 an = an−1 + 2an−2 for n ≥ 3. Find a formula for an as an = f (n). (12) Let (Fn )∞ be the Fibonacci sequence given recursively by n=0 F0 = 0 F1 = 1 Fn = Fn−1 + Fn−2 for n ≥ 2. Find a formula for Fn . What is the asymptotic behavior of log(F n )? (Here log denotes the natural logarithm.) (13) Show that A = (2, −1, 1), B = (3, 2, −1) and C = (7, 0, −2) are vertices of a right triangle. At which angle is the right angle? (14) Find a formula for the reflection of the point P = (x 0 , y0 ) about the line given by ax + by + c = 0. 2
(15) Show that the plane in R3 that passes through (x0 , y0 , z0 ) and is perpendicular to the vector (a, b, c) is given by the equation a(x − x0 ) + b(y − y0 ) + c(z − z0 ) = 0. (16) Let H = {(x, y, z) : 2x + 3y − 5z = 0}. Show that H is a subspace of R3 , and determine H ⊥ . Is H ⊥ also a subspace in of R3 ? (17) Let H be as in (15). Show that P = (1, 2, 3) is not in H and find the point P ∈ H that is closest to P . (18) For functions f, g : [a, b] → R one can define an “inner” product
b
f, g =
a
f (x)g(x) dx.
Show that this definition satisfies the four basic rules that dot product satisfies. Hence, this gives us a way to define the “norm” and “distance” between two functions and notion of two functions being perpendicular. (Note! This is used heavily in Fourier transforms and such to analyze how one can write a periodic function in terms of trigonometric ones.)
NOTE! This is not a recipe for the final exam! Geir Agnarsson April 24, 2009 3