SECTION 6.4SECTION 6.4

MATRIX MATRIX ALGEBRAALGEBRA

THE ALGEBRA OF MATRICES

THE ALGEBRA OF MATRICES

Addition:Addition:

39-

4-5

512

31-

83

1-4

PROPERTIES OF MATRICES

PROPERTIES OF MATRICES

Commutative:Commutative:

A + B = B + AA + B = B + A

Associative:Associative:

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

ZERO FOR MATRICESZERO FOR MATRICES

00

000

ADDITIVE INVERSE FOR MATRICES

ADDITIVE INVERSE FOR MATRICES

dc

ba A If

d-c-

b-a- A- then

MORE PROPERTIES OF MATRICES

MORE PROPERTIES OF MATRICES

Additive IdentityAdditive Identity:: There is a There is a matrix 0 satisfyingmatrix 0 satisfying

0 + A = A + 0 = A0 + A = A + 0 = A

Additive InversesAdditive Inverses:: For each For each matrix A, there is a matrix -A matrix A, there is a matrix -A satisfyingsatisfying

A + (-A) = (-A) + A = 0A + (-A) = (-A) + A = 0

MORE PROPERTIES OF MATRICES

MORE PROPERTIES OF MATRICES

k(A + B) = kA + kBk(A + B) = kA + kB

(k + m)A = kA + mA(k + m)A = kA + mA

(km)A = k(mA) = m(kA)(km)A = k(mA) = m(kA)

5-3-

2-7 B and

25

63- A

3A 3A = =

615

189- -2B -2B

= =

106

414-

3A - 2B = 3A - 2B =

1621

2223-

MULTIPLICATION:MULTIPLICATION:

DC

BA

dc

ba

DbC

BaA

aA+baA+bCC

bDC

aBA

aB+baB+bDD

MULTIPLICATION:MULTIPLICATION:

DC

BA

dc

ba

DdC

BcA

aA+baA+bCC

cA+dCcA+dC

dDC

cBA

aB+baB+bDD

cB+dcB+dDD

43

25- B

41-

3-2 A

Find A Find A B B

43(3)-

2-5)(2

- 19- 19

3(4)-3

)2(25-

- 8- 8

43

25- B

41-

3-2 A

44(3)

2-5)(1-

- 19- 19

Find A Find A B B

1717

4(4)3

)2(1-5-

- 8- 8

1414

Find B Find B A A

42(-1)

3-)2(5-

- 12- 12

2(4)1-

5(-3)-2-

2323

41-

3-2 A

43

25- B

44(-1)

3-)2(3

- 12- 12

22

4(4)1-

3(-3)2-

2323

77

Find B Find B A A

41-

3-2 A

43

25- B

MULTIPLICATION:MULTIPLICATION:

AB AB = =

1417

8-19-

BA BA = =

72

2312-

AB AB BA BA

MultiplicatioMultiplication is not n is not

commutativcommutative.e.

MORE PROPERTIES OF MATRICES

MORE PROPERTIES OF MATRICES

Associative:Associative:

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

Distributive:Distributive:

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

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

COMPATABILITY OF MATRICES

COMPATABILITY OF MATRICES

For Addition:For Addition: Same SizeSame Size

Counterexample:Counterexample:

241-

43-2 A

32-

25 B

COMPATABILITY OF MATRICES

COMPATABILITY OF MATRICES

For Multiplication:For Multiplication:

241-

43-2 A

32-

25 B

2-1-4

14-2

031

C

43-

1-2

2-3

D

COMPATABILITY OF MATRICES

COMPATABILITY OF MATRICES

For Multiplication:For Multiplication:

Two matrices are compatable Two matrices are compatable for multiplication when the for multiplication when the dimensions are:dimensions are: n x m and n x m and m x p m x p

The product matrix will have The product matrix will have dimension n x pdimension n x p

USING A CALCULATOR

USING A CALCULATOR

For Multiplication:For Multiplication:

241-

43-2 A

32-

25 B

2-1-4

14-2

031

C

43-

1-2

2-3

D

INVERSES OF MATRICES

INVERSES OF MATRICES

With matrices, when AB = With matrices, when AB = I I = = BA, we say A and B are BA, we say A and B are inverses.inverses.

Furthermore, when a matrix A Furthermore, when a matrix A has an inverse, we denote it as has an inverse, we denote it as AA -1 -1

Many matrices will fail to have Many matrices will fail to have an inverse.an inverse.

THEOREM: MULTIPLICATIVE

INVERSES

THEOREM: MULTIPLICATIVE

INVERSES The matrixThe matrix

dc

ba A

has a multiplicative inverse if has a multiplicative inverse if and only if D = ad - bc is and only if D = ad - bc is nonzero.nonzero.

dc

ba A

If D If D 0, 0, thenthen

Da

Dc-

Db-

Dd

A 1 -

9-5

5-2 A

Does A have an Does A have an inverse?inverse?D = 2(-9) - D = 2(-9) - 5(-5) 5(-5)

= -18 + = -18 + 25 25

= 7= 7

72

75-

75

79-

A 1 -

106-

5-3 A

Does A have an inverse?Does A have an inverse?

D = 3(10) - (-6)(-D = 3(10) - (-6)(-5) 5)

= 30 - 30 = 30 - 30

= 0= 0

INVERSES OF MATRICES

INVERSES OF MATRICES

This method of finding the This method of finding the inverse of a matrix works inverse of a matrix works nicely on 2 x 2 matrices.nicely on 2 x 2 matrices.

Finding an inverse of a 3 x 3 Finding an inverse of a 3 x 3 matrix is another matter.matrix is another matter.

We’ll use the calculator!We’ll use the calculator!

EXAMPLE:EXAMPLE:

772

672

662

A

11-0

011-

3-027

A 1 -

APPLICATIONS TO SYSTEMS OF EQUATIONS

APPLICATIONS TO SYSTEMS OF EQUATIONS

2x + 6y + 6z = 22x + 6y + 6z = 22x + 7y + 6z = - 32x + 7y + 6z = - 32x + 7y + 7z = - 52x + 7y + 7z = - 5

772

672

662

z

y

x

= =

5-

3-

2

APPLICATIONS TO SYSTEMS OF EQUATIONS

APPLICATIONS TO SYSTEMS OF EQUATIONS

772

672

662

z

y

x

= =

5-

3-

2AA- 1- 1 AA- 1- 1

z

y

x

= =

AA- 1- 1

5-

3-

2

SOLVE THE SYSTEM OF EQUATIONS

SOLVE THE SYSTEM OF EQUATIONS

2x + 6y + 6z = 22x + 6y + 6z = 22x + 7y + 6z = - 32x + 7y + 6z = - 32x + 7y + 7z = - 52x + 7y + 7z = - 5

z

y

x

= =

772

672

662

- 1- 1

5-

3-

2

SOLVE THE SYSTEM OF EQUATIONS

SOLVE THE SYSTEM OF EQUATIONS

z

y

x

= =

2-

5-

22

CONCLUSION OF SECTION 6.4CONCLUSION OF SECTION 6.4