???????? (Matrix)

(Matrix)



1.







1 2 3 1 5 6

6 8 4 2 5 6



mxn



a11 a12 a13 …a1n

a21 a22 a23 …a2n

A= . . . .

. . . .

am1 am2 am3 …amn



A =  aij  mxn

2.

A B



A=B a11 = b11 , a12 = b12



A= 1 2 B= x 2

4 6 4 y

A=B )x=1 y=6

3.

At

A



A = aijmxn At = ajinxm



4.

In nxn

1 0 0

I3 = 0 1 0

0 0 1

5.

A = aijmxn B = bijmxn

A+ B = aij + bij mxn



6.

=c

A = aijmxn

cA = caijmxn



A= 1 2 6

-1 0 3

 5A = 5 10 30

-5 0 15

7.



Amxp Bqxn = Cmxn



P q







1 2 7 1 2 9 19

3 2 8 4 5= 11 24

0 1



8. 2x2



A = a b

c d

d -b

A-1 = ad-bc ad-bc

-c a

ad-bc ad-bc



9. (Determinant)

A= a b A det(A) A

c d



det(A) = a b = ad - bc

c d

3 x3

- - -

1 2 3 1 2

4 5 6 4 5

7 8 9 7 8

+ + +

= (1)(5)(9) + (2)(6)(7) + (3)(4)(80 - (7)(5)(3)

--(8)(6)(1) - (9)(4)(2)

= 45 + 84 + 96 - 105 - 48 - 72 = 0

10. 3x3 (A-1)



1.



Mij

i j

2. (cofactor) (-1)I+j



Cij = (-1)I+j Mij



3.

aij aij

Cof.A

4. (adjoint matrix)



adj.A = (Cof.A)t

A 3x3



A-1 = 1 (adj.A) = adj.A det A  0

A detA







1. A= 1 0 1

2 1 0 A-1

1 -1 1

ฉ

1. -1/2 1/2 1/2

1 0 -1

3/2 -1/2 -1/2