كتاب المصفوفات في الرياضيات

 1 2 3  (A(  1  2 3  

 2  2 (B)  1 2   

 1 1 3    3 5 7  )C(     7 1  5

A  x  2 y  3z  0   x  2 y  3z  0

C -

-

-

( -

aij a a

a

. . .

. . . .

.

am1

a1n

. amn

n m

n

1

m

n

m=n ann

a

n×m

a

a

a

A=0

 11 2  5  A B    2  3  1  2

1 5  B  3  2

1 2  A  2  1

B A C B A A+B=B+A  A+(B+C)=(A+B)+C  k

k(A+B)=kA+kB=(A+B)k  A+D=B

2

D



a11

a12

 b11  m  b21  ... a1m    .   a11b11  a12b12  ...  a1m bm1    a1k bk1 k 1   bm1 

B B

A

A

B A

B A

A B C B A A(B+C)=AB+AC  (A+B)C=AC+BC  A(B C)=(A B)C  AB≠BA 

B=0

AB=0 

A=0 B=C

3

AB=AC 

i>j

aij=

A i<j

aij=

A=diag(a , a , a , …, ann) a = a = a = …= ann=k k=1 n

1 0  15  0  0 0

A

B

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0  0 1

1 0 0  13  0 1 0 0 0 1

1 0 12    0 1

AB=BA=1

B A B=A-

m×n AT

n×m

A 4

A

1 3 AT  2 5 2 1

k

1 2 2 A   3 5 1

BT AT

B A

A



k  AT  k  AT



A  BT  AT  BT



A  BT  AT  BT



A 

T T

A=AT

A

A A

A

A

a11 a12  a11  a22   a12  a21  a21 a22

5

a11 a12 a21 a22 a31 a32

a13 a a23  a11  22 a32 a33

a23 a a a a  a12  21 23  a13  21 22 a33 a31 a33 a31 a32

a11 a12 a21 a22 a31 a32

a13 a a23  a21  12 a32 a33

a13 a a a a  a22  11 13  a23  11 12 a33 a31 a33 a31 a32

a11 a12 a21 a22 a31 a32

a13 a a23  a31  12 a22 a33

a13 a a a a  a32  11 13  a33  11 12 a23 a21 a23 a21 a22

aij

aij

 

n

A  aij

 a11 a21 a31 a  12 ... ... adjoint A  adjA   a13 ... ...   ... ... ... a1n ... ... a b   d  b   adj( A)    A   c d   c a   b22   b32  b11 b21 b31   b   B   b12 b22 b32   adj( B)    21  b31 b   13 b23 b33   b   21  b  31

b23 b33 b23 b33 b22 b32

 

b12

b13

b32 b33 b11 b13

b31 b33 b b  11 12 b31 b32

6

 

b12 b22 b11

b21 b  11 b21

b13   b23  b13   b23  b12   b22 

... an1  ... ...  ... ...   ... ...  ... ann 

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