APP. A] VECTORS AND MATRICES 429
MISCELLANEOUS PROBLEMS
A.24. LetA=⎡
⎣100
001
110⎤
⎦andB=⎡
⎣011
100
010⎤
⎦be Boolean matrices.Find the Boolean productsAB,BA, andA^2.
Find the usual matrix product and then substitute 1 for any nonzero scalar. Thus:AB=⎡
⎣011
010
111⎤
⎦; BA=⎡
⎣111
100
001⎤
⎦; A^2 =⎡
⎣100
110
101⎤
⎦A.25. LetA=[
13
4 − 3]
.(a) Find a nonzero column vectoru=[
x
y]
such thatAu= 3 u.(b) Describe
all such vectors.(a) First set up the matrix equationAu= 3 uand then write each side as a single matrix (column vector):
[
13
4 − 3][
x
y]
= 3[
x
y]
and[
x+ 3 y
4 x− 3 y]
=[
3 x
3 y]Set corresponding elements equal to each other to obtain a system of equations, and reduce the system to echelon
form:
x+ 3 y= 3 x
4 x− 3 y= 3 y or2 x− 3 y= 0
4 x− 6 y= 0 to2 x− 3 y= 0
0 = 0
or 2x− 3 y= 0The system reduces to one (nondegenerate) linear equation in two unknowns, and so it has an infinite number
of solutions. To obtain a nonzero solution, sety=2, say; thenx=3. Thusu=[ 3 , 2 ]Tis a desired nonzero
solution.
(b) To find the general solution, sety=a, whereais a parameter. Substitutey=ainto 2x− 3 y=0 to obtain
x= 3 a/2. Thusu=[ 3 a/ 2 ,a]Trepresents all such solutions.Alternatively, lety= 2 bsov=[ 3 b, 2 b]represents
all such solutions.SupplementaryProblems
VECTORS
A.26. Letu=( 2 ,− 1 , 0 ,− 3 ), v=( 1 ,− 1 ,− 1 , 3 ), w = ( 1 , 3 ,− 2 , 2 ). Find: (a) 2u− 3 v; (b) 5u− 3 v− 4 w;
(c)−u+ 2 v− 2 w; (d)u·v, u·w, v·w, (e)‖u‖,‖v‖,‖w‖.A.27. Let u=⎡
⎣1
3
− 4⎤
⎦,v=⎡
⎣2
1
5⎤
⎦,w=⎡
⎣3
− 2
6⎤
⎦. Find: (a) 5 u − 3 v; (b) 2u + 4 v − 6 w;(c) u·v, u·w, v·w; (d) ‖u‖,‖v‖,‖w‖.A.28. Findxandywhere: (a)x( 2 , 5 )+y( 4 ,− 3 )=( 8 , 33 ); (b)x( 1 , 4 )+y( 2 ,− 5 )=( 7 , 2 ).MATRIX OPERATIONS
A.29. LetA=[
12
3 − 4]
,B=[
50
− 67]
,C=[
1 − 34
26 − 5]
,D=[
37 − 1
4 − 89]. Find:
(a) 5A− 2 Band 2C− 3 D; (c) ACandAD; (e) ATandCT;
(b) ABandBA; (d) BCandBD; (f) A^2 ,B^2 ,C^2.