AMCS 151 Tutorial 9
Question
I
find the
You
of this matrix A with an unknown x in Uentries
formula or the cofactor formula or possibly the pivotformula
determinate
could use
big
a
Find the determinant of B which has an additional 1 in thecorner
What new contribution to the determinant does this I make
X7
at
0
I
If Mis any 3 by 3 matrix
of f at a I
let fCxl dettlxm Find the derivative
Solution
Using the cofactor method we can expand
detA xl.nl det
x
ta th't detffoo
21 2 12 Hool to fool
X X M 12 213 01 10112o
x
dettal as
is 12 63 2 2 24
1,4 182 2 2 24
X4 20 2 124
It
detB
XU 202 24
2112 ol X
H 202 24 24
14 20 2
So the new contribution is 24
text detlxm
x3def M
f'txt 3 2detail
f'let 3deftM1
o
ol 1210D
it detµzF
Question Are the following statement
If A is a square matrix and def
If A
11 then A is
an orthogonal matrix
is square and A QR then IdetCall productof diagonalentriesof R
here Q is
If Q
In
A is
I P
an orthogonal
is an
Here
If
TRUE or FALSE Give brief reason
a
project
matrix and R is uppertriangular
Mxn matrix
with
orthonormal columns and m
is the mxm
identity matrix
matrix with
independent columns
onto the left nullspave NCAT
and D
n thenQQ'TIm
ACATAI AT then
Solution
false
False
counterexample
ldetla
71
example
A
if
mon
false
True
P
I P
EL
ldetlQRH
delta
ldeflQlde.HR l
I
ldetfRJ1
Ilo
then
projects
projects
CI p
A
QQT is
of
rank n
f Im
onto CCA
onto
orthogonal complement
p p p2 p_p a
of
CCA
NEAT
Questions
calculate
the
determinant
nn o o
a
L
o o
I 2
ofthe
following
4x4matrix
detlat.de
detf
I
G I
5
deflate 5
tdetf
Question
If
orthonormal
ga 92 93 are
this matrix A with
vectors in ID what're the possible determinants of
A
For
a matrix A suppose
what
2g 392and59g why
columns
29 392 593
the
information does that
cofactor
Cu of the firstentry am is zero
give about A
Can this inverse exist
find the 3 eigenvalues ofthis matrix A and find all of its eigenvectors
Why is the
diagonal'itation
a
I3 D
S AS
A not possible
Solution
dettal
Q q 92as
11 because QTQ I
detLI I
and det ata
I
detlatldettal
defeat
So when
we
determinant
multiply a column by a number the
is multiplied by the same
def I 29 392 59331
defCAI
thus
A
too
B
ff
So
whether
number
2 3 5 detLg 92 3
cofactor is zero
Cu being
the
inverse
The
eigenvalues
a a
a
too
of this
or
A
doesn't
exist
and B exists
doesn't
exists
at't
30 detCQ
130
cofactor is 7ero and
cofactor
1
give us
enough information
to decide
not
4
matrix is
EH EH
42 43 22
repeated eigenvalue
o
But the rank of A LI is 1
That means matrix A
doesn't have
A un't be diagonalitable
complete
set of
eigenvectors i e