For Exercises 61-70, use the model A = P e r t or A = P ( 1 + r n ) n t where A is the future value of P dollars invested at interest rate r compounded continuously or n times per year for t years. (See Example 11) 65. An $8000 investment grows to $9289.50 at 3% interest compounded quarterly. For how long was the money invested? Round to the nearest year.
For Exercises 61-70, use the model A = P e r t or A = P ( 1 + r n ) n t where A is the future value of P dollars invested at interest rate r compounded continuously or n times per year for t years. (See Example 11) 65. An $8000 investment grows to $9289.50 at 3% interest compounded quarterly. For how long was the money invested? Round to the nearest year.
For Exercises 61-70, use the model
A
=
P
e
r
t
or
A
=
P
(
1
+
r
n
)
n
t
where A is the future value of P dollars invested at interest rate r compounded continuously or n times per year for t years. (See Example 11)
65. An $8000 investment grows to $9289.50 at 3% interest compounded quarterly. For how long was the money invested? Round to the nearest year.
Let
2
A =
4
3
-4
0
1
(a) Show that v =
eigenvalue.
()
is an eigenvector of A and find the corresponding
(b) Find the characteristic polynomial of A and factorise it. Hint: the answer to (a)
may be useful.
(c) Determine all eigenvalues of A and find bases for the corresponding eigenspaces.
(d) Find an invertible matrix P and a diagonal matrix D such that P-¹AP = D.
(c) Let
6
0 0
A =
-10 4 8
5 1 2
(i) Find the characteristic polynomial of A and factorise it.
(ii) Determine all eigenvalues of A and find bases for the corresponding
eigenspaces.
(iii) Is A diagonalisable? Give reasons for your answer.
most 2, and let
Let P2 denote the vector space of polynomials of degree at
D: P2➡ P2
be the transformation that sends a polynomial p(t) = at² + bt+c in P2 to its derivative
p'(t)
2at+b, that is,
D(p) = p'.
(a) Prove that D is a linear transformation.
(b) Find a basis for the kernel ker(D) of the linear transformation D and compute its
nullity.
(c) Find a basis for the image im(D) of the linear transformation D and compute its
rank.
(d) Verify that the Rank-Nullity Theorem holds for the linear transformation D.
(e) Find the matrix representation of D in the standard basis (1,t, t2) of P2.
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