Expert: Sherman D. - 3/15/2005

Question
you have given  {3   1  4}    {-1  5  2}    {1 0 0}
             M={-5 -1  2}  N={8  -2 -6}  I={0 1 0}
               {2   6 -3}    {4   3  7}    {0 0 1}
find this answers:
a) IM + NI
b) MN - II
c) M(M - N)I
d) I(MM + NN)

Answer
doing a. and b., you will notice that when multiplying Matrix I, will not change anything that is in parenthesis or put into parenthesis. So what ever is done in Parenthesis will be the answer.

M =
|+3,+1,+4|
|-5,-1,+2|
|+2,+6,-3|

N =
|-1,+5,+2|
|+8,-2,-6|
|+4,+3,+7|

I =
|1,0,0|
|0,1,0|
|0,0,1|

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a.)
IM + NI
or
I(M + N)

|1,0,0|   (|+3,+1,+4| + |-1,+5,+2|)
|0,1,0| * (|+8,-2,-6| + |-5,-1,+2|)
|0,0,1|   (|+4,+3,+7| + |+2,+6,-3|)

|1,0,0|   (|(3 - 1),(1 + 5),(4 + 2)|)
|0,1,0| * (|(8 - 5),(-2 - 1),(-6 + 2|)
|0,0,1|   (|(4 + 2),(3 + 6),(7 - 3)|)

|1,0,0|   (|+2,+6,+6|)
|0,1,0| * (|+3,-3,-4|)
|0,0,1|   (|+6,+9,+4|)

|(2 + 0 + 0),(6 + 0 + 0),(6 + 0 + 0)|
|(0 + 3 + 0),(0 - 3 + 0),(0 - 4 + 0)|
|(0 + 0 + 6),(0 + 0 + 9),(0 + 0 + 4)|

ANS :
|+2,+6,+6|
|+3,-3,-4|
|+6,+9,+4|

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b.)
MN - II

(|+3,+1,+4|   |-1,+5,+2|)   |1,0,0|
(|-5,-1,+2| * |+8,-2,-6|) - |0,1,0|
(|+2,+6,-3|   |+4,+3,+7|)   |0,0,1|

(|(-3 + 8 + 16),(15 - 2 + 12),(6 - 6 + 28)|)    |1,0,0|
(|(5 - 8 + 8),(-25 + 2 + 6),(-10 + 6 + 14)|)  - |0,1,0|
(|(-2 + 48 - 12),(10 - 12 - 9),(4 - 36 - 21|)   |0,0,1|

|+21,25,+28|   |1,0,0|
|+5,-17,+10| - |0,1,0|
|34,-11,-53|   |0,0,1|

ANS :
|+20,25,+28|
|+5,-18,+10|
|34,-11,-54|

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c.) on this one, are you saying
M(M - N)I, which equals MI(M - N)

|+3,+1,+4|  (|+3,+1,+4|   |-1,+5,+2|)   |1,0,0|
|-5,-1,+2| *(|-5,-1,+2| - |+8,-2,-6|) * |0,1,0|
|+2,+6,-3|  (|+2,+6,-3|   |+4,+3,+7|)   |0,0,1|

|+3,+1,+4|   (|+4,-4,+2|)   |1,0,0|
|-5,-1,+2| * (|-13,1,+8|) * |0,1,0|
|+2,+6,-3|   (|-2,3,-10|)   |0,0,1|

|(12 - 13 - 8),(-12 + 1 + 12),(6 + 8 - 40)|
|(-20 + 13 - 4),(20 - 1 + 6),(-10 - 8 - 20)|
|(8 - 78 + 6), (-8 + 6 - 9),(4 + 48 + 30)|

ANS :
|-9,+1,-26|
|-11,25,-38|
|-64,-11,82|

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d.)
I(MM + NN)

I will shorten this, and work them on separate spaces.

|1,0,0|   (|+3,+1,+4|^2   |-1,+5,+2|^2)
|0,1,0| * (|-5,-1,+2|   + |+8,-2,-6|  )
|0,0,1|   (|+2,+6,-3|     |+4,+3,+7|  )

     |+3,+1,+4|   |+3,+1,+4|
M^2 = |-5,-1,+2| * |-5,-1,+2|
     |+2,+6,-3|   |+2,+6,-3|

|(9 - 5 + 8), (3 - 1 + 24), (12 + 2 - 12)|
|(-15 + 5 + 4),(-5 + 1 + 12),(-20 - 2 - 6)|
|(6 - 30 - 6),(2 - 6 - 18),(8 + 12 + 9)|

     |12,26,2|
M^2 = |-6,8,-28|
     |-30,-22,29|
    
     |-1,+5,+2|   |-1,+5,+2|
N^2 = |+8,-2,-6| * |+8,-2,-6|
     |+4,+3,+7|   |+4,+3,+7|

|(1 + 40 + 8),(-5 - 10 + 6),(-2 - 30 + 14)|
|(-8 - 16 - 24),(40 + 4 - 18),(16 + 12 - 42)|
|(-4 + 24 + 28),(20 - 6 + 21),(8 - 18 + 49)|

     |+49,-9,-18|
N^2 = |-48,26,-14|
     |+48,+35,39|

so this gives you

   (|12,26,+2|    |+49,-9,-18|)
I * (|-6,+8,-28| - |-48,26,-14|)
   (|-30,-22,29|  |+48,+35,39|)

   (|-37,+35,+20|)
I * (|+42,-18,-14|)
   (|-78,-57,-10|)

ANS :
|-37,+35,+20|
|+42,-18,-14|
|-78,-57,-10|

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for Matrix Rules, go to http://easyweb.easynet.co.uk/~mrmeanie/matrix/matrices.htm

For a matrix Calculator, go to www.quickmath.com and click on Arithmetic under Matrices

If you use quickmath, leave out the | or your problem will not show up. Just use the values that you have inside the brackets and make sure to put commas between them.

By the way, if you are wondering why i put + instead of just leaving it like it is, i was hoping to make them line up. Also to let you know, sorry if the problems get pushed over to the left, i have no control over that, since at this site, that is what happens when we send them.

Also when you go from subtracting the matrices to multpling them. Make sure not to forget to change how the math is preformed.

I hope i was helpful in answering your problem.

Also since it would take up to much space, i just went ahead and worked out the multipling. Instead of doing
(1 * 2) + (3 * 4) + (5 * 6), i just went ahead and did 2 + 12 + 30.

To see how i did it, just use the first site i gave you.