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~A ~B ˆ a11 B ~ !
a 12
~B
ˆ
a 21
~B a 22
~B
MATRIX ALGEBRA
0
1
a 11 b 11 a 11 b 12 a 12 b 11 a 12 b 12
a 11 b 21 a 11 b 22 a 12 b 21 a 12 b 22
B a 21 b 11 a 21 b 12 a 22 b 11 a 22 b 12
C
@
A :
a 21 b 21 a 21 b 22 a 22 b 21 a 22 b 22
Problems
3.1 For the pairs A ~ and ~B given below, ®nd A ~ ‡ ~B, A ~ ~B, and A ~ 2 :
~A ˆ 1 2
; ~B ˆ 5 6
:
3 4
7 8
3.2 Show that an n-rowed diagonal matrix ~D
0
1
k 0 0
0 k 0
~D ˆ
B
.
.
. . C
@
A
commutes with any n-rowed square matrix ~A: ~A ~D ˆ ~D ~A ˆ k ~A.
3.3 If ~A, ~B, and ~C are any matrices such that the addition ~B ‡ ~C and the
products ~A ~B and ~A ~C are de®ned, show that ~A( ~B ‡ ~C† ˆ ~A ~B + ~A ~C. That
is, that matrix multiplication is distributive.
3.4 Given
0 1 0 1 0 1
0 1 0
1 0 0
1 0 0
B C B C B C
~A ˆ @ 1 0 1A; ~B ˆ @ 0 1 0A; ~C ˆ @ 0 0 0A;
0 1 0
0 0 1
0 0 1
show that [ ~ A; ~BŠ ˆ0, and [ ~B; ~ CŠˆ0, but that ~ A does not commute with ~ C.
3.5 Prove that ( ~A ‡ ~B† T ˆ ~A T ‡ ~B T .
3.6 Given
~A ˆ 2 3
; ~B ˆ 5 2
; and ~C ˆ 0 1 2
:
0 4
2 1
3 0 4
(a) Find 2 ~A 4 ~B, 2(~A 2 ~B)
(b) Find ~A T ; ~B T ; … ~B T † T
(c) Find ~C T ; … ~C T † T
(d) Is ~A ‡ ~C de®ned?
(e) Is ~C ‡ ~C T de®ned?
( f ) Is ~ A ‡ ~ A T symmetric?
140
k