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1 (Matrix and Vector)
(1)
(2)
• 2
3x + y = 4
9x − 3y = 8
=⇒
3 1
9 −3
x
y
=
4
8
3
•
a11x1 + a12x2 + · · · + a1nxn = d1
a21x1 + a22x2 + · · · + a2nxn = d2
...
...
...
...
am1x1 + am2x2 + · · · + amnxn = dm
(a) x
(b)
(c)
A = x = d =
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• A
A = (aij) (i = 1, 2, · · · , m, j = 1, 2, · · · , n)
(row) (column)
• A = (aij) i j
• × aij
• m = n
(3)
• (column vector): ::::::
1
• (row vector): ::::::
1
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•
1
1
n
⇓
n n– n
n– 1
• x d
• x d
•
x
2×2
=
x11 x12
x21 x22
=⇒
d
m×1
=
d1
d2
...
dm
=⇒
(4)
1 Ax = d 2
(a) 2 A x
(b) Ax d
1
⇓
(Matrix Algebra)
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(1) pp.65–69
(2) 4.1 (p.69) 1 2
2
(1)
A = (aij) B = (bij)
A = B ⇐⇒ aij = bij for all i and j.
4 3
0 2
=
4 3
0 2
=
3 4
0 2
x
y
=
7
4
x = y =
(2)
•
•
3 8
9 5
2×2
+
4 1
2 7
2×2
=
a11 a12
a21 a22
a31 a32
3×2
−
b11 b12
b21 b22
b31 b32
3×2
=
(3)
=⇒
7
3 −1
0 5
=
1
2
a11 a12
a21 a22
=
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3
(1)
u =
4
2
5
v =
2
1
2
u v =
uv =
(2)
2 n n
(a) u =
3
2
or u = ( 3 2 )
u u (0, 0) u or u 2
(3, 2) 1 u
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• ( 1 0 ) ( 0 1 ) i 1 0
(b) 3 3 3
3 p.87 4.4
(c) n- n- n-
n- 1 n n
(1) pp.79–88
(2) 4.3 1–6 (p.89)
4
a b A B
: a + b = b + a
: ab = ba
: (a + b) + c = a + (b + c)
: (ab)c = a(bc)
: a(b + c) = ab + ac
(1) pp.90–93
(2) 4.4 1–5
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5
(1) (Identity Matrices)
1
I
2×2
= I
3×3
=
I
4×4
=
• 1
1 × a = a × 1 = a
IA = AI = A
A
2×3
=
1 2 3
2 0 3
I
2×2
A
2×3
=
A
2×3
I
3×3
=
• A
m×n
I
n×n
B
n×p
=
• (I)2
=
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(2) (Null Matrices)
O =
0 0 0
0 0 0
• A
m×n
+ O
m×n
= O
m×n
+ A
m×n
= A
m×n
• A
m×n
O
n×p
= O
m×p
O
q×m
A
m×n
= O
q×n
(3)
• ab = 0 a b
AB =
2 4
1 2
−2 4
1 −2
=
• cd = ce d = e c = 0
C =
2 3
6 9
, D =
1 1
1 2
, E =
−2 1
3 2
CD = CE =
(1) pp.94–96
(2) 4.5 1–3
6
(1) (transposed matrices)
A =
2 5
4 9
A = AT
=
B
m×n
B n×m
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(2)
(A ) =
(A + B) =
(AB) =
(3) (inverse matrices)
• A
• AA−1
= A−1
A =
• A A
⇐⇒ A
⇐⇒ A
•
(A−1
)−1
= A
• A n × n A−1
n × n
•
• (AB)−1
=
• (A )−1
=
II 16