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NAME ______________________________________________ DATE______________________________ PERIOD _____________

". [  6 10

−4 3

−2 7 ]  ⋅  [0 4 –3] #. [7   −25   −4 ]  ⋅  [   1   −3−2 0 ] $. [

  2 0   −3

1 4   −2

−1 3 1 ]  ⋅

[

  2   −23 1

−2 4

 ]

C!pt"r # 38 Glencoe Algebra

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3-6 Study Guide and Intervention (continued)Multiplying Matrices

Multiplicati%e &roperties he !ommutati"e #roperty of $ultiplication does not hold for matrices.

Properties of Matrix Multiplication $or !ny m!tric"s A% B% !n& C for wic t" m!tri' pro&uct is&"fin"&% !n& !ny sc!l!r c % t" followin( prop"rti"s !r" tru".

Associative Property of Matrix Multiplication ) AB*C = A)BC *

Associative Property of Scalar Multiplication c ) AB* = )cA*B = A)cB*

Left Distributive Property C ) A + B* = CA + CB

Right Distributive Property ) A + B*C = AC + BC 

Example: 'se A = [4   −32 1 ] , B = [2 05   −3] , and C = [1   −26 3 ]  to find each product.a. ( A ) B*C 

  % A & B' C = ([4   −32 1 ]+[2 05   −3])  ⋅  [1   −26 3 ]= [6   −37   −2]  ⋅ [1   −26 3 ]= [6(1)+(−3)(6)   6(−2)+(−3)(3)7(1)+(−2)(6)   7(−2)+(−2)(3)]=

[−12   −21

−5   −20]b. AC ) BC 

  AC & BC = [4   −32 1 ]  ⋅  [1   −26 3 ]  & [2 05   −3]  ⋅  [1   −26 3 ]= [4 (1)+(−3)(6)   4 (−2)+(−3)(3)2 ( 1 )+1 (6 )   2 (−2 )+1 ( 3 )   ]  ) [   2 ( 1 )+0 (6 )   2 (−2 )+0 ( 3 )5 (1)+(−3)(6)   5(−2)+(−3)(3)]=

[

−14   −17

8   −1

 ] &

[

  2   −4

−13   −19

] =

[

−12   −21

−5   −20

] (ote that althou)h the results in the example illustrate the *i)ht +istributi"e #roperty, they do not pro"e it.Exercises

'se A = [3 25   −2] , B = [6 42 1] , C = [−1

2−2

1   −3] , and scalar c = +4 to determine hether the folloin-

euations are true for the -i%en matrices.

C!pt"r # 38 Glencoe Algebra

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1. c% AB' = %cA' B 2. AB = BA

3. BC = CB 4. % AB'C = A% BC '

. C % A & B' = AC & BC  !. c% A & B' = cA & cB

C!pt"r # 38 Glencoe Algebra