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2J - Se ond Midterm
Notes, b o oks and al ulators are not allowed. Explain all your answer.
Undetailed answer will not re eived full redit
1 Exer ise 1 (25 p oints)
Let us onsider the following matrix:
A =
A
Find the eigenvalues and orresp onding eigenve tors.
Solution: The hara teristi p olynomial of A is:
p() = det(A I )
The eigenvalues are 1
= 1 and 3
= 2. The orresp onding eigenve tors are given by
(A
i
)v i
= 0 (with i = 1 ; 2 ; 3). This gives v 1
A
, v 2
A
, v 3
A
Let A b e the following matrix:
A =
- ( 10 p oints) Find the eigenvalues and eigenve tors of A. Can A b e diagonalized?
- ( 10 p oints) Find A
- ( 10 p oints) Without any omputation, an you explain why the matrix A do es not have
an inverse?
Solution: The hara teristi p olynomial of the matrix A is p() = ( + 1). There ex-
ist 2 eigenvalues 1 = 0 and 2 = 1. The orresp onding eigenve tors are v 1
and
v 1
. The matrix A is a n n matrix and we have n eigenve tors, therefore the matrix
an b e diagonalized (another way to say this is to noti e that, the 2 eigenvalues have multipli ity
1 and b oth have 1 eigenve tor).
Let us de ne the matrix X = (v 1
; v 2
. Therefore X
. We
an then write A as A = X D X
with D the diagonalized matrix D =
This gives that A
= X D
X
. D b eing diagonal, we have D
. We an
see that D
= D. Then A
= X D
X
= X D X
, that is A
= A.
We an easily on lude that A an not have an inverse, b e ause = 0 is an eigenvalue.
- ( 10 p oints) What are the 3rd order Taylor approximation of e
x
and sin(x) ab out = 0?
- ( 15 p oints) What is the 3rd order Taylor approximation of f (x) =
e
x
1 sin(x)
ab out = 0?
Solution: The 3rd order approximation of e
x
is e
x
' 1 + x +
x
x
, and sin(x) ' x
x
(Also 3! = 6).
The Taylor approximation of
1 x
is
1 x
' 1 + x + x
. Repla ing x by sin(x), we get
1 sin(x)
' 1 + sin(x) + sin
(x) + sin
(x)
' 1 + (x
x
) + (x
x
x
' 1 + x + x
x
Remark: in the al ulation ab ove, we dropp ed the x
, x
,... terms sin e we are only interested
in the 3rd order approximation.
Then
e
x
1 sin(x)
= e
x
1 sin(x)
' e
x
(1 + sin(x) + sin
(x) + sin
(x))
' (1 + x +
x
x
)(1 + x + x
x
' 1 + 2 x +
x
