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Young Won Lim3/13/18
Matrix (2A)
Young Won Lim3/13/18
Copyright (c) 2015 - 2018 Young W. Lim.
Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License".
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Functions (4A) 3 Young Won Lim3/13/18
Row Vector, Column Vector, Square Matrix
https://en.wikipedia.org/wiki/Matrix_(mathematics)
Functions (4A) 4 Young Won Lim3/13/18
Matrix Notation
https://en.wikipedia.org/wiki/Matrix_(mathematics)
Functions (4A) 5 Young Won Lim3/13/18
Matrix Addition
https://en.wikipedia.org/wiki/Matrix_(mathematics)
Functions (4A) 6 Young Won Lim3/13/18
Scalar Multiplication
https://en.wikipedia.org/wiki/Matrix_(mathematics)
Functions (4A) 7 Young Won Lim3/13/18
Transposition
https://en.wikipedia.org/wiki/Matrix_(mathematics)
Functions (4A) 8 Young Won Lim3/13/18
Matrix Multiplication
https://en.wikipedia.org/wiki/Matrix_(mathematics)
Functions (4A) 9 Young Won Lim3/13/18
Properties
https://en.wikipedia.org/wiki/Matrix_(mathematics)
Functions (4A) 10 Young Won Lim3/13/18
Matrix Types
https://en.wikipedia.org/wiki/Matrix_(mathematics)
Functions (4A) 11 Young Won Lim3/13/18
Identity Matrix
https://en.wikipedia.org/wiki/Matrix_(mathematics)
Functions (4A) 12 Young Won Lim3/13/18
Inverse Matrix
https://en.wikipedia.org/wiki/Invertible_matrix
Functions (4A) 13 Young Won Lim3/13/18
Inverse Matrix Examples
https://en.wikipedia.org/wiki/Matrix_(mathematics)
Functions (4A) 14 Young Won Lim3/13/18
Solving Linear Equations
A set of linear equations
[a bc d ][ xy ] = [ef ]
a x + b y = e
c x + d y = f [ xy ] = [a bc d]
−1
[ef ]
∣a bc d∣ = ad − bc ≠ 0
∣e bf d∣
[ xy ] =1
ad − bc [ d −b−c a ][ef ]
∣a ec f∣
x =∣e bf d∣
∣a bc d∣
y =∣a ec f∣
∣a bc d∣
= de − bf
= af − ce
[ xy ] =1
ad − bc [ de−bf−ce+af ]
Cramer's Rule
If the inverse matrix exists
Determinant (3A) 15 Young Won Lim03/13/2018
Rule of Sarrus (1)
[a11 a12 a13
a21 a22 a23
a31 a32 a33]
a11 a12
a21 a22
a31 a32[a11 a12 a13
a21 a22 a23
a31 a32 a33]
a11 a12
a21 a22
a31 a32
Determinant of order 3 only
Rule of Sarrus
[a11 a12 a13
a21 a22 a23
a31 a32 a33] [
a11 a12 a13
a21 a22 a23
a31 a32 a33]
a11 a12
a21 a22
a31 a32
Copy and concatenate
+ + + – – –
+ a11a22a33 + a12a23a31 + a13a21a32 − a13a22a31 − a11a23 a32 − a12a21a33
Determinant (3A) 16 Young Won Lim03/13/2018
Linear Equations
a11 x1 + a12 x2 + a1n xn = b1+⋯
a21 x1 + a22 x2 + a2n xn = b2+⋯
an1 x1 + an2 x2 + ann xn = bn+⋯
⋮ ⋮ ⋮ ⋮
a11 a12 a1n
a21 a22 a2n
an1 an2 ann
⋮ ⋮⋮
⋯
⋯
⋯
x1
x2
xn
⋮
= b1
= b2
= bn
⋮
(Eq 1)
(Eq 2)
(Eq 3)
Ax = b
Determinant (3A) 17 Young Won Lim03/13/2018
Cramer's Rule (1) – solutions
a11 a12 a1n
a21 a22 a2n
an1 an2 ann
⋮ ⋮⋮
⋯
⋯
⋯
x1
x2
xn
⋮
= b1
= b2
= bn
⋮
A
a11 a12 a1n
a21 a22 a2n
am1 an2 ann
⋮ ⋮⋮
⋯
⋯
⋯
a11 a12a1n
a21 a22a2n
an1 am2ann
⋮ ⋮⋮⋯
⋯
⋯
a11 a12 a1n
a21 a22 a2n
an1 an2 amn
⋮ ⋮⋮
⋯
⋯
⋯
b1
b2
bn
⋮
b1
b2
bn
⋮
b1
b2
bn
⋮
x b
A2A1 An
x1 =det (A1)
det (A)x2 =
det (A2)
det (A)xn =
det (An)
det (A)
Young Won Lim3/13/18
References
[1] http://en.wikipedia.org/[2]