QUADRATIC FUNCTIONS

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SECTION 6.4
MATRIX ALGEBRA
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THE ALGEBRA OF MATRICES
Addition
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PROPERTIES OF MATRICES
Commutative A B B A Associative A
(B C) (A B) C
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ZERO FOR MATRICES
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ADDITIVE INVERSE FOR MATRICES
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MORE PROPERTIES OF MATRICES
Additive Identity There is a matrix 0
satisfying 0 A A 0 A Additive
Inverses For each matrix A, there is a matrix
-A satisfying A (-A) (-A) A 0
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MORE PROPERTIES OF MATRICES
k(A B) kA kB (k m)A kA mA (km)A
k(mA) m(kA)
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3A
-2B
3A - 2B
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MULTIPLICATION
aAbC
aBbD
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MULTIPLICATION

aAbC
aBbD
cAdC
cBdD
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Find A B
- 19
- 8
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Find A B

- 19
- 8
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14
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Find B A
- 12
23
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Find B A

- 12
23
2
7
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MULTIPLICATION
AB
AB ¹ BA

Multiplication is not commutative.
BA
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MORE PROPERTIES OF MATRICES
Associative A (B C) (A B) C
Distributive A (B C) A B A C (B
C) A B A C A
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COMPATABILITY OF MATRICES
For Addition Same Size Counterexample
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COMPATABILITY OF MATRICES
For Multiplication
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COMPATABILITY OF MATRICES
For Multiplication Two matrices are compatable
for multiplication when the dimensions are n x m
and m x p The product matrix will have
dimension n x p
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USING A CALCULATOR
For Multiplication
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INVERSES OF MATRICES
With matrices, when AB I BA, we say A and B
are inverses. Furthermore, when a matrix A has an
inverse, we denote it as A -1 Many matrices will
fail to have an inverse.
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THEOREM MULTIPLICATIVE INVERSES
The matrix
has a multiplicative inverse if and only if D
ad - bc is nonzero.
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If D ¹ 0, then
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Does A have an inverse?
D 2(-9) - 5(-5) -18 25 7
YES!
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Does A have an inverse?
D 3(10) - (-6)(-5) 30 - 30
0
NO!
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INVERSES OF MATRICES
This method of finding the inverse of a matrix
works nicely on 2 x 2 matrices. Finding an
inverse of a 3 x 3 matrix is another
matter. Well use the calculator!
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EXAMPLE
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APPLICATIONS TO SYSTEMS OF EQUATIONS
2x 6y 6z 2
2x 7y 6z - 3
2x 7y 7z - 5

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APPLICATIONS TO SYSTEMS OF EQUATIONS
A- 1

A- 1
A- 1

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SOLVE THE SYSTEM OF EQUATIONS
2x 6y 6z 2
2x 7y 6z - 3
2x 7y 7z - 5
- 1

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SOLVE THE SYSTEM OF EQUATIONS

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• CONCLUSION OF SECTION 6.4