chapter 1, what can't be ignored: section 1.1 to addition and multiplication review of matrices
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7/23/2019 Chapter 1, What Can't Be Ignored: Section 1.1 To Addition And Multiplication Review Of Matrices
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R
. . .
Z
mn
π
√ 2
7/23/2019 Chapter 1, What Can't Be Ignored: Section 1.1 To Addition And Multiplication Review Of Matrices
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F
R
R
aij
a11 a12 . . . a1na21 a22 . . . a2n
am1 am2 . . . amn
= A = ( aij)
∈ Rmxn
( aij) + ( bij)
( a23)
a11 a12 a13a21 a22 a23
7/23/2019 Chapter 1, What Can't Be Ignored: Section 1.1 To Addition And Multiplication Review Of Matrices
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1 2 34 5 6
( b23) b11 b12 b13b21 b22 b23
7 8 9
10 11 12
( aij + bij)
1 2 34 5 6
7 8 9
10 11 12
(a11 + b11) (a12 + b12) (a13 + b13)(a21 + b21) (a22 + b22) (a23 + b23)
(1 + 7) (2 + 8) (3 + 9)
(4 + 10) (5 + 11) (6 + 12)
8 10 12
14 16 18
a11 a12 . . . a1 p
am1 am2 . . . amp
·
b11 b12 . . . b1nb21 b22 . . . b2n
b p1 b p2 . . . b pn
c11 . . . c1n
cm1 . . . cmn
cij =
pk=1
aikbkj , for i = 1, . . . , m, j = 1, . . . , n .
7/23/2019 Chapter 1, What Can't Be Ignored: Section 1.1 To Addition And Multiplication Review Of Matrices
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a11 a12 . . . a1 pa21 a22 . . . a2 p
am1 am2 . . . amp
·
x1x2
x p
c1c2
cm
cij=1 =
pk=1
aikbk1 =
pk=1
aikxk
1 2 34 5 6
A = (aij) = A(i, j) =
a11 a12 a13a21 a22 a23
1 2 34 5 6
. . . . . .
B = (bij) = B(i, j) =
b11 b12 b13b21 b22 b23
7 8 9
10 11 12
7/23/2019 Chapter 1, What Can't Be Ignored: Section 1.1 To Addition And Multiplication Review Of Matrices
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C = (cij) = C (i, j) =
c11 c12
c21 c22c31 c32
13 14
15 1617 18
·
dij =
pk=1
aikckj , for i = 1, . . . , m, j = 1, . . . , n .
D = (dij) = D(i, j) = A · C =
1 2 34 5 6
·13 14
15 1617 18
94 100
229 244
7/23/2019 Chapter 1, What Can't Be Ignored: Section 1.1 To Addition And Multiplication Review Of Matrices
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= 2 = 3 AC 23 is (row2) · (column3) = p
k=1 a2kck3
·
A(4x5)·C (5x6) =
. . .
ai1 ai2 · · · ai5. . .
. . .
·
. . . . . . c1j . . . . . . . . .
c2j
c5j
. . .
. . . . . . (AC ij) . . . . . . . .. . .
. . .
D = (dij) = D(i, j) = A · C =
1 2 34 5 6
·13 14
15 1617 18
94 100
229 244
dij
d11 = 94
c1 = c(:, 1)
c1 = (ci1)
13
1517
r1 = A(1, :)
r1 = (a1i) =
1 2 3
7/23/2019 Chapter 1, What Can't Be Ignored: Section 1.1 To Addition And Multiplication Review Of Matrices
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d11 = (a1i) · (ci1) =
a11 a12 a13 · c11
c21c31
1 2 3
·13
1517
d11 =
3
k=1 a1kbk1 (a11 · c11) + (a12 · c21) + ( a13 · c31) = (1)(13) + (2)(15) + (3)(17) =13 + 30 + 51 = 94
d12 (a1i)
r1 = A(1, :)
r1 = (a1i) =
1 2 3
ci2
c2 = C (:, 2)
c2 = (ci2)
14
1618
d12 = r1 · c2 = (a1i) · (ci2) =
a11 a12 a13 · c12
c22c32
1 2 3
·14
1618
d12 =
3
k=1 a1k ck2 = (a11 · c12) + (a12+c22) + (a13+ c32) = (1)(14) + (2)(16) + (3)(18) =
d21 d22
dij = p
k=1 aikckj
=
7/23/2019 Chapter 1, What Can't Be Ignored: Section 1.1 To Addition And Multiplication Review Of Matrices
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