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Solve Linear Equations by Matrix Methods

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2 Answers

8.8        -1.5         1.6

3.3         3.0          -.1               =102.4

5.5         4.5         3.2

 

13.32         -1.5         1.6

  1.24          3.0        -.1           =61.51

18.76          4.5        3.2

 

8.8         13.32         1.6

3.3           1.24         -.1           =-8.42

5.5         18.76         3.2

 

8.8         -1.5         13.32

3.3          3.0           1.24             =506.8

5.5          4.5         18.76

Do the arithmetic, and  x=0.60     y=-0.082     z=4.9

Use the coefficients in each equation and set up a matrix: [ 8.8 -1.5 1.6 ] [ x ] [ 13.32 ] [ 3.3 3.0 -0.1 ] [ y ] = [ 1.24 ] [ 5.5 4.5 3.2 ] [ z ] [ 18.76 ] By doing arithmetic on the rows of the matrix (e.g., adding one row to another) we wish to get to a final solution of the form: [ 1.0 0.0 0.0 ] [ x ] [ ??.?? ] [ 0.0 1.0 0.0 ] [ y ] = [ ??.?? ] [ 0.0 0.0 1.0 ] [ z ] [ ??.?? ] When we have achieved that, we'll have the correct answer in the matrix on the right. I'll use E1, E2 and E3 as shorthand to represent each of the three equations, in order. If you subtract E2 from E1, the result is [ 8.8 -1.5 1.6 ] [ 13.32 ] -[ 3.3 3.0 -0.1 ] = [ 1.24 ] ====================================== [ 5.5 -4.5 1.7 ] [ 12.08 ] Now our matrix is [ 5.5 -4.5 1.7 ] [ x ] [ 12.08 ] [ 3.3 3.0 -0.1 ] [ y ] = [ 1.24 ] [ 5.5 4.5 3.2 ] [ z ] [ 18.76 ] Subtracting E1 from E3: [ 5.5 -4.5 1.7 ] [ x ] [ 12.08 ] [ 3.3 3.0 -0.1 ] [ y ] = [ 1.24 ] [ 0.0 9.0 1.5 ] [ z ] [ 6.68 ] We have made progress-one of the terms is now zero! There's still lots more work to do... but this will get you started.

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