Algebra 2 Matrix Operations
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Son of the Matrix
Matrix Operations

 

 Now let's look at a group of 3 equations with three variables ...

 
 Example:
 Solve these using matrix algebra ...
 
  2X + 5Y - Z = 21
  5X - Y + 4Z = 31
  2X - 3Y - 5Z = -31
 
 The way we solve this one starts out the same, but bigger.
 

 

 
 The way we combine these is a bit different.
 To explain it, we're going to use a matrix with just letters in it.
 

 

 
 Now we combine the smaller 4 number (letters here) matrices (plural of matrix)
 the way we did on the last page ...
 

 D = A ((E I) - (H F)) - B ((D I) - (G F)) + C ((D H) - (G E))

 
 Don't even think about memorizing this equation
 just remember how to find it.
 
 The matrices with 4 numbers in them 
 are always done the same way.
 
 There are many ways that the bigger 9 number matrices
 can be worked out.
 
 This one is as good as any other.
 
 1)    Cross out the top row, then cross out the left column.
   What's left is a matrix with 4 numbers in it.
   The 4 numbers get multiplied together
   and then multiplied by the number 
   that was crossed out twice in the big matrix.
   
2)   Cross out the top row, then cross out the middle column.
   What's left is a matrix with 4 numbers in it.
   This gets multiplied together
   and then multiplied by the number 
   that was crossed out twice in the big matrix.
   This value will be subtracted from the other two values.
   
3)    Cross out the top row, then cross out the right column.
   What's left is a matrix with 4 numbers in it.
   This gets multiplied together
   and then multiplied by the number 
   that was crossed out twice in the big matrix.
   
4)    Add the first and third number and subtract the second.
   THIS IS THE ANSWER

  

 Let's get back to our problem ...
 

 

 
 So:
  D = 2((-1(-5))-((-3)4))-5((5(-5))-(24))+(-1)((5(-3))-(2(-1))
  D = 2((5)-(-12))-5((-25)-(8))+(-1)((-15)-(-2))
  D = 2(17)-5(-33)+(-1)(-13)
  D = 34 + 165 + 13
  D = 212
 
 Now we find Dx by substituting the answers for the X coefficients
 and doing the same process all over again
 

 

 
  Dx = 21((-1(-5))-((-3)4))-(-5)((31(-5))-(-314))+(-1)((31(-3))-(-31(-1))
  Dx = 21((5)-(-12))-(-5)((-155)-(-124))+(-1)((-93)-(31))
  Dx = 21(17)-(-5)(-31)+(-1)(-124)
  Dx = 357 + 155 + 124
  Dx = 636
 
 
 Now find Dy by substituting the answers for the Y coefficients ...
 

 

 
  DY = 2((31(-5))-((-31)4))-(21)((5(-5))-(24))+(-1)((5(-31))-(2(31))
  DY = 2((-155)-(-124))-(21)((-25)-(8))+(-1)((-155)-(62))
  DY = 2(-31)-(21)(-33)+(-1)(-217)
  DY = - 62 + 693 + 217
  DY = 848
 
 
 Now we need to find Z.
 Once we have X and Y, we can use substitution to find Z.
 In fact, once we have ONE value,
 we can use substitution to find the others.
 But just for practice or because it's so much fun (yeah right),
 lets use matrices to find Z ...
 

 

 
  DZ = 2((-1(-31))-((-3)31))-(5)((5(-31))-(231))+(21)((5(-3))-(2(-1))
  DZ = 2((31)-(-93))-(5)((-155)-(62))+(21)((-15)-(-2))
  DZ = 2(124)-(5)(-217)+(21)(-13)
  DZ = 248 + 1085 - 273
  DZ = 1060
 
 
 As you can see, 
 the chances of making simple math errors on this stuff is really big.
 Computers don't make that kind of mistake
 so they love this stuff!
 

   copyright 2005 Bruce Kirkpatrick

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