IES Alfonso X El Sabio Murcia Linear ALGEBRA

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http://www.vitutor.com/alg/matrix/matrices_index.html

1 Matrices

A m a t r i x i s e v e r y s e t o f n u m b e r s o r t e r m s a r r a n g e d i n a

r e c t a n g u l a r s h a p e , f o r m i n g r o w s a n d c o l u m n s .

E a c h n u m b e r i n a m a t r i x i s a n e l e m e n t. O n e e l e m e n t i s d i s t i n g u i s h e d f r o m a n o t h e r b y i t s p o s i t i o n , t h a t i s t o s a y , t h e r o w

a n d c o l u m n t o w h i c h i t b e l o n g s .

T h e n u m b e r o f r o w s a n d c o l u m n s o f a m a t r i x i s c a l l e d t h e

d i m e n s i o n o f a m a t r i x. T h u s , a m a t r i x i s o f d i m e n s i o n : 2 x 4 , 3 x 2 , 2 x 5 , . . . I f t h e m a t r i x h a s t h e s a m e n u m b e r o f r o w s a n d c o l u m n s , i s

s a i d t o b e o f o r d e r : 2 , 3 , . . .

T h e s e t o f m a t r i c e s o f m r o w s a n d n c o l u m n s i s d e n o t e d b y

Am x n o r ( ai j), a n d a n y e l e m e n t w i t h i n t h e m a t r i x i s i n r o w i i n

c o l u m n j, f o r ai j.

T w o m a t r i c e s a r e e q u a l w h e n t h e y h a v e t h e s a m e d i m e n s i o n a n d

e q u a l e l e m e n t s w h i c h o c c u p y t h e s a m e p l a c e i n b o t h .

R o w M a t r i x

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2

C o l u m n M a t r i x

A c o l u m n m a t r i x i s f o r m e d b y a s i n g l e c o l u m n .

R e c t a n g u l a r M a t r i x

A r e c t a n g u l a r m a t r i x i s f o r m e d b y a d i f f e r e n t n u m b e r o f r o w s a n d c o l u m n s , a n d i t s d i m e n s i o n i s n o t e d a s : m x n.

S q u a r e M a t r i x

A s q u a r e m a t r i x i s f o r m e d b y t h e s a m e n u m b e r o f r o w s a n d c o l u m n s .

T h e e l e m e n t s o f t h e f o r m ai i c o n s t i t u t e t h e p r i n c i p a l d i a g o n a l .

T h e s e c o n d a r y d i a g o n a l i s f o r m e d b y t h e e l e m e n t s w i t h i + j = n + 1.

Z e r o M a t r i x

I n a z e r o m a t r i x , a l l t h e e l e m e n t s a r e z e r o s .

U p p e r T r i a n g u l a r M a t r i x

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3

L o w e r T r i a n g u l a r M a t r i x

I n a l o w e r t r i a n g u l a r m a t r i x , t h e e l e m e n t s a b o v e t h e d i a g o n a l a r e z e r o s .

D i a g o n a l M a t r i x

I n a d i a g o n a l m a t r i x , a l l t h e e l e m e n t s a b o v e a n d b e l o w t h e d i a g o n a l a r e z e r o s .

S c a l a r M a t r i x

A s c a l a r m a t r i x i s a d i a g o n a l m a t r i x i n w h i c h t h e d i a g o n a l e l e m e n t s a r e e q u a l .

I d e n t i t y M a t r i x

A n i d e n t i t y m a t r i x i s a d i a g o n a l m a t r i x i n w h i c h t h e d i a g o n a l e l e m e n t s a r e e q u a l t o 1 .

T r a n s p o s e M a t r i x

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4

( A + B )t = At + Bt ( α A )t_{= α A}t

( A B )t = Bt At

R e g u l a r M a t r i x

A r e g u l a r m a t r i x i s a s q u a r e m a t r i x t h a t h a s a n i n v e r s e .

S i n g u l a r M a t r i x

A s i n g u l a r m a t r i x i s a s q u a r e m a t r i x t h a t h a s n o i n v e r s e .

I d e m p o t e n t M a t r i x

T h e m a t r i x A i s i d e m p o t e n t i f :

A2 = A.

I n v o l u t i v e M a t r i x

T h e m a t r i x A i s i n v o l u t i v e i f :

A2_{= I}_.

S y m m e t r i c M a t r i x

A s y m m e t r i c m a t r i x i s a s q u a r e m a t r i x t h a t v e r i f i e s :

A = At_.

A n t i s y m m e t r i c M a t r i x

A n a n t i s y m m e t r i c m a t r i x i s a s q u a r e m a t r i x t h a t v e r i f i e s :

A = − At.

O r t h o g o n a l M a t r i x

A m a t r i x i s o r t h o g o n a l i f i t v e r i f i e s t h a t :

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5 A d d i n g M a tr i c e s

G i v e n t w o m a t r i c e s o f t h e s a m e d i m e n s i o n , A = ( ai j) a n d B = ( bi j), t h e

m a t r i x s u m i s d e f i n e d a s : A + B = ( ai j + bi j). T h a t i s , t h e r e s u l t a n t

m a t r i x ' s e l e m e n t s a r e o b t a i n e d b y a d d i n g t h e e l e m e n t s o f t h e t w o m a t r i c e s t h a t o c c u p y t h e s a m e p o s i t i o n .

P r o p e r ti e s o f t h e A d d it i o n of Ma t r i c e s

C l o s u r e :

T h e s u m o f t w o m a t r i c e s o f d i m e n s i o n m x n i s a n o t h e r m a t i x o f d i m e n s i o n m x n .

A s s o c i a t i v e :

A + ( B + C ) = ( A + B ) + C

A d d i t i v e i d e n t i t y :

A + 0 = A

W h e r e 0 i s t h e z e r o m a t r i x o f t h e s a m e d i m e n s i o n .

A d d i t i v e i n v e r s e :

A + ( − A ) = O

T h e o p p o s i t e m a t r i x h a s e a c h o f i t s e l e m e n t s c h a n g e s i g n .

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6

E x a m p l e s

G i v e n t h e m a t r i c e s :

C a l c u l a t e :

A + B ; A - B

S c a l a r M a t r i x Mu l t i p l i c a t i o n

G i v e n a m a t r i x , A = ( ai j), a n d a r e a l n u m b e r , k R, t h e p r o d u c t o f a r e a l

n u m b e r b y a m a t r i x i s a m a t r i x o f t h e s a m e d i m e n s i o n a s A, a n d e a c h e l e m e n t i s m u l t i p l i e d b y k.

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7 P r o p e r ti e s

a ( b A ) = ( a b ) A A Mm x n, a , b

a ( A + B ) = a A + a B A , B Mm x n , a

( a + b ) A = a A + b A A Mm x n , a , b

1 A = A A Mm x n

M u l t i p l y i n g M a tr i c e s

T w o m a t r i c e s A a n d B c a n b e m u l i t p l i e d t o g e t h e r i f t h e n u m b e r o f c o l u m n s o f A i s e q u a l t o t h e n u m b e r o f r o w s o f B .

Mm x n x Mn x p = Mm x p

T h e e l e m e n t , ci j, o f t h e p r o d u c t m a t r i x i s o b t a i n e d b y m u l t i p l y i n g e v e r y

e l e m e n t i n r o w i o f m a t r i x A_{b y e a c h e l e m e n t o f c o l u m n}_j_{o f m a t r i x}B_{a n d}

t h e n a d d i n g t h e m t o g e t h e r .

P r o p e r ti e s o f Ma t r i x M u lt i p l i c a ti o n

A s s o c i a t i v e :

A ( B C ) = ( A B ) C

M u l t i p l i c a t i v e I d e n t i t y

A I = A

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8

A B ≠ B A

D i s t r i b u t i v e :

A ( B + C ) = A B + A C

E x a m p l e s

1 .

G i v e n t h e m a t r i c e s :

C a l c u l a t e :

A x B ; B x A

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9

D e t e r m i n e i f t h e f o l l o w i n g m u l t i p l i c a t i o n s a r e p o s s i b l e :

1 .

( At B ) C

( At3 x 2 B2 x 2) C3 x 2 = ( At B )3 x 2 C3 x 2

T h e m u l t i p l i c a t i o n i s n o t p o s s i b l e b e c a u s e t h e n u m b e r o f c o l u m n s , ( At B ) d o e s n o t c o i n c i d e w i t h t h e n u m b e r s o f r o w s o f C .

2 .

( B Ct _{) A}t

( B2 x 2 C t2 x 3 ) At3 x 2 = ( B C )2 x 3 At3 x 2 =

= ( B C t A t ) 2 x 2

3D e t e r m i n e t h e d i m e n s i o n o f

M s o t h a t t h e m u l t i p l i c a t i o n i s p o s s i b l e : A M C

A3 x 2 Mm x n C3 x 2 m = 2

4D e t e r m i n e t h e d i m e n s i o n o f

M i f C t M i s a s q u a r e m a t r i x . C t2 x 3 Mm x n m = 3 n = 3

M a t r i x I n v e r se

T h e m u l t i p l i c a t i o n o f a m a t r i x b y i t s i n v e r s e i s e q u a l t o t h e i d e n t i t y m a t r i x .

A A- 1 = A- 1 A = I

P r o p e r t i e s o f t h e I n v e r s e M a t r i x

( A B )- 1 = B- 1 A- 1 ( A- 1₎- 1_{= A}

( k A )- 1_{= k}- 1_A- 1

( A t)- 1 = ( A - 1)t

S t e p s t o C a l c u l a t e t h e I n v e r s e M a t r i x

A i s a s q u a r e m a t r i x o f o r d e r n . T o c a l c u l a t e t h e i n v e r s e o f A , d e n o t e d a s

A- 1, f o l l o w t h e s e s t e p s :

1 C o n s t r u c t a m a t r i x o f t y p e

M = ( A | I ), t h a t i s t o s a y , A_{i s i n t h e l e f t}

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10

P l a c e t h e i d e n t i t y m a t r i x o f o r d e r 3 t o t h e r i g h t o f M a r t i x M_.

2U s i n g t h e

G a u s s i a n e l i m i n a t i o n m e t h o d, t r a n s f o r m t h e l e f t h a l f , A, t o t h e i d e n t i t y m a t r i x , l o c a t e d t o t h e r i g h t , a n d t h e m a t i x t h a t r e s u l t s i n t h e r i g h t s i d e w i l l b e t h e i n v e r s e o f m a t r i x : A- 1_.

r2 - r1

r3 + r2

r2 - r3

r1 + r2

( − 1 ) r2

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11

E x a m p l e s

C a l c u l a t e t h e m a t r i x i n v e r s e o f :

1 C o n s t r u c t a m a t r i x o f t y p e M = ( A | I ) .

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F o r w h a t v a l u e s o f m i n t h e m a t r i x

d o e s n o t s u p p o r t a n i n v e r s e ?

F o r a n y r e a l v a l u e o f m , t h e r e i s t h e i n v e r s e A- 1_.

C a l c u l a t i n g t h e m a t r i x i n v e r s e f o r d e t e r m i n a t s

C a l c u l a t i n g t h e M a t r i x I n v e r s e f o r D e t e r m i n a n t s

S t e p s t o C a l c u l a t e t h e M a t r i x I n v e r s e

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13 2 .

T h e a d j u g a t e m a t r i x c a n b e f o u n d b y r e p l a c i n g e v e r y e l e m e n t b y i t s

c o f a c t o r.

3 .

C a l c u l a t e t h e t r a n s p o s e o f t h e a d j u g a t e m a t r i x.

4 .

T h e m a t r i x i n v e r s e i s e q u a l t o t h e i n v e r s e v a l u e o f i t s d e t e r m i n a n t m u l t i p l i e d b y t h e t r a n s p o s e o f t h e a d j u g a t e m a t r i x .

T h e m a t r i x i n v e r s e c a n a l s o b e c a l c u l a t e d b y t h e G a u s s i a n e l i m i n a t i o n m e t h o d .

E x a m p l e s

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14 2 .

F o r w h a t v a l u e s o f x i n t h e m a t r i x d o e s t h e m a t r i x i n v e r s e n o t s u p p o r t ?

F o r x = 0 , M a t r i x A h a s n o i n v e r s e .

R a n k o f a M a tr ix

T h e r a n k o f a m a t r i x i s t h e n u m b e r o f l i n e s i n t h e m a t r i x ( r o w s o r c o l u m n s ) t h a t a r e l i n e a r l y i n d e p e n d e n t .

A l i n e i s l i n e a r l y d e p e n d e n t o n a n o t h e r o n e o r o t h e r s w h e n a l i n e a r c o m b i n a t i o n b e t w e e n t h e m c a n b e e s t a b l i s h e d .

A l i n e i s l i n e a r l y i n d e p e n d e n t o f a n o t h e r o n e o r o t h e r s w h e n a l i n e a r c o m b i n a t i o n b e t w e e n t h e m c a n n o t b e e s t a b l i s h e d .

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C a l c u l a t i n g t h e R a n k o f a M a t r i x

T h e G a u s s i a n e l i m i n a t i o n m e t h o d i s u s e d t o c a l c u l a t e t h e r a n k o f a m a t r i x .

A l i n e c a n b e d i s c a r d e d i f :

• A l l t h e c o e f f i c i e n t s a r e z e r o s .

• T h e r e a r e t w o e q u a l l i n e s .

• A l i n e i s p r o p o r t i o n a l t o a n o t h e r .

• A l i n e i s a l i n e a r c o m b i n a t i o n o f o t h e r s .

r3 = 2 r1

r4 i s z e r o

r5 = 2 r2 + r1

r ( A ) = 2 .

I n g e n e r a l , e l i m i n a t e t h e m a x i m u m p o s s i b l e n u m b e r o f l i n e s , a n d t h e r a n g e i s t h e n u m b e r o f n o n z e r o r o w s .

r2 = r2 − 3 r1

r3= r3 − 2 r1

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16

E x a m p l e

C a l c u l a t e t h e r a n k o f t h e f o l l o w i n g m a t r i x :

r1 − 2 r2

r3 − 3 r2

r3 + 2 r1

T h e r e f o r e , r ( A ) = 2 .

T h e r a n k o f a m a t r i x i s t h e n u m b e r o f l i n e a r l y i n d e p e n d e n t r o w s o r c o l u m n s . U s i n g t h i s d e f i n i t i o n , t h e r a n k c a n b e c a l c u l a t e d u s i n g t h e G a u s s i a n e l i m i n a t i o n m e t h o d.

I t c a n a l s o b e s a i d t h a t t h e r a n k i s : t h e o r d e r o f t h e l a r g e s t n o n z e r o s q u a r e s u b m a t r i x . U s i n g t h i s d e f i n i t i o n , t h e r a n k c a n b e c a l c u l a t e d u s i n g d e t e r m i n a n t s .

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17 1 . A l i n e c a n e l i m i n a t e d a i f :

A l l t h e c o e f f i c i e n t s a r e z e r o s .

T h e r e a r e t w o e q u a l l i n e s .

A l i n e i s p r o p o r t i o n a l t o a n o t h e r .

A l i n e i s a l i n e a r c o m b i n a t i o n o f o t h e r s .

T h e t h i r d c o l u m n c a n b e d e l e t e d b e c a u s e i t i s a l i n e a r c o m b i n a t i o n o f t h e f i r s t t w o : c3 = c1 + c2

2 .

C h e c k t o s e e i f t h e r a n k i s 1 , f o r i t m u s t b e s a t i s f i e d t h a t t h e e l e m e n t o f t h e m a t r i x i s n o t z e r o a n d t h e r e f o r e i t s d e t e r m i n a n t i s n o t z e r o .

| 2 | = 2 ≠ 0

3 .

T h e m a t r i x w i l l h a v e a r a n k o f 2 i f t h e r e i s a s q u a r e s u b m a t r i x o f o r d e r 2 a n d i t s d e t e r m i n a n t i s n o t z e r o .

4 .

T h e m a t r i x w i l l h a v e a r a n k o f 3 i f t h e r e i s a s q u a r e s u b m a t r i x o f o r d e r 3 a n d i t s d e t e r m i n a n t i s n o t z e r o .

A s a l l t h e d e t e r m i n a n t s o f t h e s u b m a t r i c e s a r e z e r o , i t d o e s n o t h a v e a r a n k o f 3 , t h e r e f o r e r ( B ) = 2.

I f t h e m a t r i x h a d a r a n k o f 3 a n d t h e r e w a s a s u b m a t r i x o f o r d e r 4 , w h o s e d e t e r m i n a n t w a s n o t z e r o , i t w o u l d h a v e h a d a r a n k o f 4 . I n t h e s a m e w a y a s s h o w n a b o v e , c h e c k t o s e e i f t h e r e i s a r a n g e g r e a t e r t h a n 4 .

E x a m p l e s

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18

| 2 | = 2 ≠ 0

r ( A ) = 2

2 .

C a l c u l a t e t h e r a n k o f t h e m a t r i x :

r ( B ) = 4

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19

R e m o v e t h e t h i r d c o l u m n a s i t i s z e r o , t h e f o u r t h b e c a u s e i t i s p r o p o r t i o n a l t o t h e f i r s t a n d t h e f i f t h b e c a u s e i t i s t h e l i n e a r c o m b i n a t i o n o f t h e f i r s t a n d s e c o n d : c5 = − 2 c1 + c2

r ( C ) = 2

D e t er m i n a n t s

E v e r y s q u a r e m a t r i x, A, i s a s s i g n e d a p a r t i c u l a r s c a l a r q u a n t i t y c a l l e d t h e

d e t e r m i n a n t o f A, d e n o t e d b y | A |o r b y d e t ( A ).

A =

D e t e r m i n a n t o f O r d e r O n e

| a1 1| = a1 1

| 5 | = 5

D e t e r m i n a n t o f O r d e r T w o

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20

D e t e r m i n a n t o f O r d e r T h r e e

C o n s i d e r a n a r b i t r a r y 3 x 3 m a t r i x , A = ( ai j) . T h e d e t e r m i n a n t

o f A i s d e f i n e d a s f o l l o w s :

=

a1 1 a2 2 a3 3 + a1 2 a2 3 a 3 1 + a1 3 a2 1 a3 2 -

- a 1 3 a2 2 a3 1 - a1 2 a2 1 a 3 3 - a1 1 a2 3 a3 2 .

=

3 2 4 + 2 ( - 5 ) ( - 2 ) + 1 0 1 -

- 1 2 ( - 2 ) - 2 0 4 - 3 ( - 5 ) 1 =

= 2 4 + 2 0 + 0 - ( - 4 ) - 0 - ( - 1 5 ) =

= 4 4 + 4 + 1 5 = 6 3

N o t e t h a t t h e r e a r e s i x p r o d u c t s , e a c h c o n s i s t i n g o f t h r e e e l e m e n t s i n t h e m a t r i x . T h r e e o f t h e p r o d u c t s a p p e a r w i t h a p o s i t i v e s i g n ( t h e y p r e s e r v e t h e i r s i g n ) a n d t h r e e w i t h a n e g a t i v e s i g n ( t h e y c h a n g e t h e i r s i g n ) .

W e c a n s o l v e a 3 x 3 d e t e r m i n a n t b y a p p l y i n g t h e f o l l o w i n g f o r m u l a :

=

a1 1 a2 2 a3 3 + a1 2 a2 3 a 3 1 + a1 3 a2 1 a3 2 -

- a 1 3 a2 2 a3 1 - a1 2 a2 1 a 3 3 - a1 1 a2 3 a3 2 .

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21 R u l e o f S a r r u s

T h e t e r m s w i t h a + s i g n a r e f o r m e d b y t h e e l e m e n t s o f t h e p r i n c i p a l d i a g o n a l a n d t h o s e o f t h e p a r a l l e l d i a g o n a l s w i t h i t s c o r r e s p o n d i n g

o p p o s i t e v e r t e x.

T h e t e r m s w i t h a − s i g n a r e f o r m e d b y t h e e l e m e n t s o f t h e s e c o n d a r y d i a g o n a l a n d t h o s e o f t h e p a r a l l e l d i a g o n a l s w i t h i t s c o r r e s p o n d i n g

o p p o s i t e v e r t e x.

E x a m p l e

M i n o r a n d C o f a ct o r

M i n o r

A n e l e m e n t , ai j, t o t h e v a l u e o f t h e d e t e r m i n a n t o f o r d e r n − 1 , o b t a i n e d

b y d e l e t i n g t h e r o w i a n d t h e c o l u m n j i n t h e m a t r i x i s c a l l e d a m i n o r.

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22 C o f a c t o r

T h e c o f a c t o r o f t h e e l e m e n t ai j i s i t s m i n o r p r e f i x i n g :

T h e + s i g n i f i + j i s e v e n.

T h e − s i g n i f i + j i s o d d.

T h e v a l u e o f a d e t e r m i n a n t i s e q u a l t o t h e s u m o f t h e p r o d u c t s o f t h e e l e m e n t s o f a l i n e b y i t s c o r r e s p o n d i n g c o f a c t o r s :

E x a m p l e

= 3 ( 8 + 5 ) - 2 ( 0 - 1 0 ) + 1 ( 0 + 4 ) = 3 9 + 2 0 + 4 = 6 3

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23 1 .

| At_{| = | A |}

T h e d e t e r m i n a n t o f m a t r i x A a n d i t s t r a n s p o s e At_{a r e e q u a l .}

2 .

| A | = 0 I f :

I t h a s t w o e q u a l l i n e s

A l l e l e m e n t s o f a l i n e a r e z e r o .

T h e e l e m e n t s o f a l i n e a r e a l i n e a r c o m b i n a t i o n o f t h e o t h e r s .

r3 = r1 + r2

3 .

A t r i a n g u l a r d e t e r m i n a n t i s t h e p r o d u c t o f t h e d i a g o n a l e l e m e n t s .

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24 5 .

I f t h e e l e m e n t s o f a l i n e a r e a d d e d t o t h e e l e m e n t s o f a n o t h e r p a r a l l e l l i n e p r e v i o u s l y m u l t i p l i e d b y a r e a l n u m b e r , t h e v a l u e o f t h e d e t e r m i n a n t i s u n c h a n g e d .

6 .

I f a d e t e r m i n a n t i s m u l t i p l i e d b y a r e a l n u m b e r , a n y l i n e c a n b e m u l t i p l i e d b y t h e a b o v e m e n t i o n e d n u m b e r , b u t o n l y o n e .

7 .

I f a l l t h e e l e m e n t s o f a l i n e o r c o l u m n a r e f o r m e d b y t w o a d d e n d s , t h e a b o v e m e n t i o n e d d e t e r m i n a n t d e c o m p o s e s i n t h e s u m o f t w o d e t e r m i n a n t s .

8 .

| A B | = | A | | B |

T h e d e t e r m i n a n t o f a p r o d u c t e q u a l s t h e p r o d u c t o f t h e d e t e r m i n a n t s .

E x a m p l e s

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25

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26 3 .

C a l c u l a t e :

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27 4 x 4 D e t er m i n a n t

T o s o l v e a d e t e r m i n a n t o f o r d e r 4 o r h i g h e r , o n e o f t h e l i n e s o f t h e d e t e r m i n a n t s h o u l d b e f o r m e d b y z e r o s , e x c e p t o n e : t h e b a s e e l e m e n t w h i c h w i l l b e w o r t h 1 o r − 1 .

F o l l o w t h e s e s t e p s :

1 .

I f a n y e l e m e n t o f t h e d e t e r m i n a n t i s 1 , c h o o s e o n e o f t h e f o l l o w i n g l i n e s : t h e r o w o r c o l u m n c o n t a i n i n g t h e 1 . K e e p i n m i n d , t h a t t h e l i n e t h a t c o n t a i n s t h e l a r g e s t n u m b e r o f z e r o e l e m e n t s s h o u l d b e s e l e c t e d .

2 .

I f t h e r e i s n o 1 i n t h e e l e m e n t s :

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28 2 .

D i v i d e t h e l i n e b y o n e o f i t s e l e m e n t s , t h e n t h e d e t e r m i n a n t s h o u l d b e m u l t i p l i e d b y t h a t e l e m e n t s o t h a t i t s v a l u e d o e s n o t v a r y .

3 .

C o n t i n u e t o o p e r a t e u n t i l a l l t h e e l e m e n t s o f t h e r o w o r c o l u m n w h e r e t h e b a s i c e l e m e n t i s l o c a t e d a r e z e r o s .

4 .

T a k e t h e c o f a c t o r o f t h e b a s i c e l e m e n t , s o t h a t a d e t e r m i n a n t o f o r d e r o n e l e s s t h a n t h e o r i g i n a l i s o b t a i n e d .

= 2 ( - 5 8 )

V a n d e r m o n d e D e t e rm i n a n t

A V a n d e r m o n d e d e t e r m i n a n t p r e s e n t s a g e o m e t r i c s e q u e n c e i n e v e r y r o w o r i n e v e r y c o l u m n w i t h t h e f i r s t e l e m e n t b e i n g 1 .

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29

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30 Linear Systems

Linear Equation with n Unknowns

I t is any exp ressio n such as: a1x1 + a2x2 + a3x3 + ... + anxn = b, where ai, b .

Where, ai, are the co efficients, b, the ind ep endent term and xi , the unkno wns. Solut ion of a Linear Eq uat ion

Any set o f n real numb ers that verifies the eq uatio n is called a so lutio n to the eq uatio n.

Given the eq uatio n x + y + z + t = 0, the solutio ns to it are:

( 1, −1, 1, −1), ( −2, −2, 0, 4). Eq uivalent Eq uat ions

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31 Systems of Linear Equations

I t is a set o f algebraic expressions in the fo rm :

a1 1x1 + a1 2x2 + ... ... .... ... .... ... .+a1 nxn = b1

a2 1 x 1 + a2 2x2 + . ... ... .... ... .... ...+a2 nxn = b2

. .... ... .... ... ... .... ... .... ... .... ... .... .... ... .... ... ... ...

am 1x1 + am 2x2 + . .... ... .... ... .... .. +am nxn = bm

• xi are the unkno wns, ( i = 1, 2, . .., n).

• ai j are the co efficients, (i = 1, 2, . .., m), ( j = 1, 2, ... , n) . • bi are the ind ep endent terms, ( i = 1, 2, . .., m) .

• m, n ; m > n, o r, m = n, o r, m < n.

• No te that the numb er of eq uatio ns need no t eq ual the numb er o f unkno wns. • ai j and bi .

• When n takes a lo w value, it is usual to d esig nate the unkno wns with the letters x, y, z, t, ...

• When bi = 0, for all i, the system is called homogeneous.

Solut ion of a Syst em

I t is each set of values that satisfies all eq uations.

Equivalent Systems of Equations

Eq uivalent equatio n systems have the sam e solutio n, altho ug h they m ay have a d ifferent numb ers of eq uatio ns.

Eq uivalent system s o f equatio ns are o b tained b y eliminatio n if:

All co efficients are zeros.

Two ro ws are eq ual.

A ro w is p ro po rtio nal to ano ther.

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32 Equivalence Criteria

1 I f b o th memb ers o f an eq uatio n of a system are added o r subtracted b y the same exp ressio n, the resulting system is eq uivalent.

2I f bo th m emb ers o f the eq uatio ns o f a system are m ultiplied o r d ivid ed b y a numb er o ther than zero, the resultant system is eq uivalent.

3I f an equatio n of a system is ad d ed or red uced by ano ther eq uatio n o f the same system , the resultant system is eq uivalent.

4I f an equatio n in a system is replaced b y ano ther eq uatio n that results fro m add ing the equatio ns o f a system p reviously m ultip lied or divid ed by no nzero num b ers, the resultant system is eq uivalent.

5 I f the o rd er o f the eq uations o r the o rder of the unkno wns of a system is chang ed, it is ano ther eq uivalent system.

Classifying Systems of Linear Equations

Consid ering t he N umb er of its Solut ions

Inc onsistent

No so lution

Consistent

I t has a so lution.

Consistent ind ep endent

I t has a single solutio n.

Consistent d ep end ent

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33 System of Linear Equations in Triangular

Form

They are a system of equatio ns that have an unkno wn less in each eq uatio n than the eq uatio n p revio us.

x + y + z = 3 y + 2 z = −1

z = −1

I n the 3rd eq uatio n, there is z = −1.

Sub stituting this value into the 2nd eq uatio n, it becom es y = 1.

And sub stituting this into the 1st equatio n, it becom es x = 3.

x + y + z = 4 y + z = 2

With this system, there are mo re unkno wns than there are equatio ns. I n this case, take o ne o f the unknowns ( eg z) and change its m em b er.

x + y + z = 3 y = 2 − z

C o nsid er z = λ , with λ b eing a p arameter to take any real value.

x + y + z = 3 y = 2 − λ

The solutio ns are:

z= λ y = 2 − λ x= 1.

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34

The Gauss eliminatio n m etho d is to transfo rm a system of equatio ns into an eq uivalent system that is in triang ular form.

To facilitate the calculatio n, transform the system into a m atrix and place the co efficients of the variab les and the ind ep end ent term s into the m atrix ( sep arated b y a straig ht line).

Examp le:

3x +2y + z = 1

5x +3y +4z = 2

x + y - z = 1

C hange the po sitio n o f ro w 3 to where ro w 1 is and shift the o ther ro ws do wn acco rdingly:

Sub tract ro w 2 b y three tim es the value of ro w 1 and sub tract ro w 3 by five tim es the value of ro w 1.

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36

Examp les:

1

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37

3

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38

5

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39 Cramer's Rule

C ramer' s rule is used to solve system s of linear equatio ns. I t ap plies to systems that m eet the fo llo wing co nd itio ns:

The num b er o f eq uations eq uals the numb er o f unkno wns.

The d eterm inant of the co efficient m atrix is no nzero .

∆ is the d eterminant o f the co efficient m atrix.

And they are:

∆ 1, ∆ 2 , ∆ 3 ... , ∆ n

D eterminants are ob tained b y rep lacing the co efficients of the 2nd m em b er ( ind ep end ent term s) in the 1st, 2nd, 3rd and the nth colum n, resp ectively.

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40

Examp les

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41

2

3

Solving Systems of Equations

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42

• r = r' Consist ent syst em.

o r = r' = n Consistent independ ent syst em.

o r = r'≠ n Consistent d ep endent system. • r ≠ r' Inconsist ent system.

St ep s t o Solve a Syst em of Eq uat ions

1. Find the rank of t he mat rix o f coefficients:

r(A) = 3

2. Find the rank of the augm ented m atrix:

r(A') = 3

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43

4. So lve the system if it is no t inco nsistent, b y Cramer's rule or the Gauss

eliminat ion met hod.

Take the system co rrespo nd ing to the sub m atrix o f ord er 3, which has a rank o f 3 and so lve it:

Examp les

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45

3.

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46 Homogeneous Systems

I f a system o f m eq uatio ns and n unkno wns has all zero ind ep end ent term s, it is said to b e homogeneous.

I t o nly ad mits the trivial so lution:x1 = x2 =... = xn = 0.

The necessary and sufficient co nditio n fo r a ho mog eneo us system has solutio ns o ther than the trivial when the rank of the coefficient m atrix is less than the num b er o f unkno wns, that is to say, that the determ inant of the co efficient m atrix is zero.

r < n

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47

r = 3 n = 3

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48

1. Solving a Linear Syst em of Eq uat ions wit h P aramet ers b y Cramer's Rule

1. Find the rank of the m atrix o f co efficients:

2. Find the rank of the augm ented m atrix:

3. Study the ob tained inform atio n and d etermine what typ e o f system it is:

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49

2. Solving a Linear Syst em of Eq uat ions wit h P aramet ers b y t he Gauss Eliminat ion Method

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50

Examp les

If a = 0.

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