APP. A] VECTORS AND MATRICES 431
MISCELLANEOUS PROBLEMS
A.45. LetA=[
12
01]
.Find: (a)An; (b)A−^1 ; (c) matrixBsuch thatB^2 =A.A.46. MatricesAandBare said to commute ifAB=BA. Find all matrices[
xy
zt]
which commute with[
11
01]
.A.47. LetA=⎡
⎣010
101
100⎤
⎦andB=⎡
⎣100
100
011⎤
⎦be Boolean matrices.Find the Boolean matrices: (a)A+B; (b)AB; (c)BA; (d)A^2 ; (e)B^2.Answers to Supplementary Problems
Notation:M=[R 1 ;R 2 ;...;Rn]denotes a matrix with
rowsR 1 ,...,Rn.
A.26. (a)( 1 , 1 , 3 ,− 15 ); (b)( 3 ,− 14 , 11 ,− 32 );
(c)(− 2 ,− 7 , 2 , 5 ); (d)− 6 ,− 7 ,6;
(e)√
14 ,√
12 = 2√
3 ,√
18 = 3√
2.
A.27. (a)[− 1 , 12 ,− 35 ]T; (b)[− 8 , 22 ,− 24 ]T;
(c)− 15 ,− 27 ,34; (d)√
26 ,√
30 ,7.
A.28. (a)x= 6 ,y=−1; (b)x= 3 ,y=2.
A.29. (a)[− 5 , 10 ; 27 ,− 34 ],[− 7 , 27 , 11 ;− 8 , 36 ,− 37 ];
(b)[− 7 , 14 ; 39 ,− 28 ],[ 5 , 10 ; 15 ,− 40 ];
(c)[ 5 , 9 ,− 6 ;− 5 ,− 33 , 32 ],[ 11 ,− 9 , 17 ;− 7 , 53 , 39 ];
(d)[ 5 ,− 15 , 20 ; 8 , 60 ,− 59 ],[ 15 , 35 ,− 5 ; 10 ,− 98 , 69 ];
(e)[ 1 , 3 ; 2 ,− 4 ],[ 1 , 2 ;− 3 , 6 ; 4 ,− 5 ];
(f)[ 7 ,− 6 ;− 9 , 22 ],[ 25 , 0 ;− 72 , 49 ],C^2 not defined.
A.30. (a)[− 13 ,− 3 , 18 ; 4 , 17 , 0 ]; (b)ABnot defined,
[− 5 ,− 2 , 4 , 5 ; 11 ,− 3 ,− 12 , 18 ],[ 9 ; 9 ];
(c)[ 11 ,− 12 , 0 ,− 5 ;− 15 , 5 , 8 , 4 ],[− 1 ; 9 ],CDnot
defined; (d)[ 1 , 0 ;− 1 , 3 ; 2 , 4 ],[ 4 , 0 ,− 3 ;− 7 ,
− 6 , 12 ; 4 ,− 8 , 6 ].
A.31. [ 2 , 4 , 6 ;− 1 ,− 2 ,− 3 ]
A.32. (a)[ 2 ,− 6 ,− 1 ],−5;(b)[ 1 , 2 ,− 1 ], 2
A.33. (a)[− 11 ,− 15 ; 9 ,− 14 ],[− 67 , 40 ;− 24 ,− 59 ];
(b)[− 50 , 70 ;− 42 ,− 36 ].
A.34. (a)[ 14 , 4 ;− 2 , 34 ],[ 60 ,− 52 ; 26 ,− 200 ]
(b)f(B)=0.A.35. [2a;a], for any nonzeroa.
A.36. (a)−18; (b)−15; (c) 44; (d)−b^2.
A.37. (a) 323; (b) 48.
A.38. (a)[ 3 ,− 4 ;− 5 , 7 ]; (b)[ 1 ,− 2 / 3 ; 2 ,− 5 / 3 ];
(c) Not defined.
A.39. (a)[− 16 ,− 11 , 3 ; 7 / 2 , 5 / 2 ,− 1 / 2 ;− 5 / 2 ,− 3 / 2 ,
1/2]; (b)[ 1 , 1 / 2 , 0 ;− 1 / 2 ,− 1 / 2 , 1 / 2 ;− 1 / 2 ,− 1 ,
1/2]; (c) Not defined.
A.40. (a)[ 1 , 2 ,− 1 , 2 , 1 ; 0 , 0 , 3 ,− 6 , 1 ; 0 , 0 , 0 ,− 6 , 1 ],
[ 1 , 2 , 0 , 0 , 4 / 3 ; 0 , 0 , 1 , 0 , 0 ; 0 , 0 , 0 , 1 ,− 1 / 6 ];
(b)[ 2 , 3 ,− 2 , 5 , 1 ; 0 ,− 11 , 10 ,− 15 , 5 ; 0 ,..., 0 ],
[ 1 , 0 , 4 / 11 , 5 / 11 , 13 / 11 ; 0 , 1 ,− 10 / 11 , 15 / 11 ,
− 5 / 11 ; 0 ,..., 0 ]
A.41.[ 1 , 1 ; 0 , 1 ],[ 1 , 1 ; 0 , 0 ],[ 1 , 0 ; 0 , 0 ],[ 0 , 1 ; 0 , 0 ],
[ 0 , 0 ; 0 , 0 ],[ 1 , 0 ; 0 , 1 ]
A.42.Thereare13.
A.43. (a)x= 3 ,y= 1 ,z=2;(b)No solution.
A.44. (a)x= 3 y+ 5 t,z= 1 − 2 t; (b)x= 2 ,y= 1 ,z=1.
A.45. (a)[ 1 , 2 n; 0 , 1 ]; (b)[ 1 ,− 2 ; 0 , 1 ]; (c)[ 1 , 1 ; 0 , 1 ].
A.46. [a, b; 0 ,a]
A.47. (a)[ 110 ; 101 ; 111 ]; (b)[ 100 ; 111 ; 100 ];
(c)[ 010 ; 010 ; 101 ]; (d)[ 101 ; 110 ; 010 ];
(e)[ 100 ; 100 ; 111 ].