16
A
REVIEW OF
VECTOR
AND
MATRIX ALGEBRA
With matrix notation, a product between these two matrices can be expressed as
[kfI[Nl.
Using subscript/summation notation,
this
same product is expressed as
(a)
With the elements of the
[MJ
and
[N]
matrices given above, evaluate the
matrix products
[MJ[Nl,
[N[MJ
and
[MI[kfl,
leaving the results
in
matrix
array notation.
(b)
Express the matrix products
of
part (a) using the subscriptlsummation nota-
tion.
(c)
Convert the following expressions, which
are
written in subscriptlsummation
notation, to
matrix
notation:
Mi
jNjk.
i.
MjkNij.
ii.
MijNkj.
fi.
M
..N.
Jl
Jk*
iV.
MijNjk
$.
Tki.
V.
MjiNkj
-k
Tik.
2.
Consider the square
3
X
3 matrix
[MI
whose elements
Mij
are
generated by the
expression
M..
11
=
ij2
and
a
vector
v
whose components
in
some basis are given by
vk
=
k
forc,
j
=
1,2,3,
fork
=
1,2,3.
(a)
Using a matrix representation for
-+
[u,
determine the components of
(b)
Determine the components of the vector that result from a premultiplication
the vector that result from a premultiplication of
[u
by
[MI.
of
[MI
by
[Ut.
3.
Thematrix
1
,Rl
=
[
cos8 sin8
-sin8 cos8
represents a rotation. Show that the matrices
[RI2
=
[R][R]
and
[RI3
=
[RlER][Rl
are matrices that correspond to rotations of
28
and
38
respectively.
4.
Let
[D]
be a 2
X
2 square matrix and
[V]
a 2
X
1
row
matrix. Determine the
conditions imposed on
[D]
by the requirement that
[D"
=
[VI+[Dl
for
any
[V].
5.
The trace of
a
matrix
is
the sum of all its diagonal elements. Using the sub-
scriptlsummation notation to represent the elements
of
the matrices
[TI
and
lM1,
