Young Won Lim3/13/18

Matrix (2A)

Young Won Lim3/13/18

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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]