MATH 152 REVIEW PROBLEMS → R (1) Suppose that T : R

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MATH 152 REVIEW PROBLEMS
(1) Suppose that T : R2 → R3 is a linear transformation satisfying
 
 
2
1
 
T   = −1
 
4
3
 
 
0
2
 
T = 4 
 
6
−2
Find T [ 01 ].
(2) Given the transition matrix


3
4
1
4
3
4
1
4
,
find the equilibrium probability.
(3) Find the determinant of the matrix


0 1
7 5


0 0

0
6



.
0 0 −3 3


2 −1 4 0
(4) Find the inverse of


1 2 3


0 1 2 .


0 0 1
(5) Let A be given by the matrix


25 0 0 


 0 35 0  .


0 0 30
(a) What is det A?
(b) What are the eigenvalues of A?
(c) What is A−1 ?
(6) Find all vectors in R4 that are orthogonal to (1, 2, −2, 3), (−2, 3, 5, −1), and (−4, 13, 11, 3).
1
2
MATH 152 REVIEW PROBLEMS
(7) If A has eigenvalues and eigenvectors given by
λ1 = 1
k~1 = (1, 0, 5)T
λ2 = 0
k~2 = (0, 1, 1)T
λ3 =
k~3 = (0, 1, −1)T
1
2
(a) Write (1, 0, 0)T as a linear combination of eigenvectors.
(b) What is A5 (1, 0, 0)T ?
~
(c) What is the solution X(t)
to the system of linear differential equations given by
~
dX
~
= AX
dt
~
with X(0)
= (1, 0, 0)T ?
(8) Consider the system given by
d~x
= A~x
dt
(1)
where A has eigenvalues and eigenvectors
λ1 = 1
k~1 = (1, 0, 1)T
λ2 = 2 + i
k~2 = (i, 1, 0)T
λ3 = 2 − i
k~3 = (−i, 1, 0)T .
(a) Write the general solution to (1).
(b) Find the solutions of (1) (written in terms of real functions of t) that satisfy the initial
conditions
~x(0) = (1, 2, 3)T .