Solving a linear system with matrices using Gaussian elimination - Matrices

Solving a linear system with matrices using Gaussian elimination

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Notes:
Note
Gaussian elimination (or row reduction) is a method used for solving linear systems. For example,

x+y+z=3x+y+z=3
x+2y+3z=0x+2y+3z=0
x+3y+2z=3x+3y+2z=3

Can be represented as the matrix:



Using Gaussian elimination, we can turn this matrix into

(watch the intro video to learn how to do this!)

Now we can start solving for x,yx,y and zz.

So in the third row, we see that 3z=6-3z=6. So z=2z=-2.

In the second row, we see that 2y+4z=62y+4z=-6. Since we know that z=2z=-2, then we can substitute it into the second row and solve for yy. So,

2y+4z=62y+4z=-6 2y+4(2)=6 2y+4(-2)=-6
2y8=6 2y-8=-6
2y=2 2y=2
y=1 y=1

So now we know that z=2z=-2, and y=1y=1. Now let us take a look at the first row and solve for xx.

x+y+z=3x+y+z=3 x+12=3 x+1-2=3
x1=3 x-1=3
x=4 x=4

Since we have solved for x,yx,y and zz, then we have just solved the linear system.
  • 2.
    Gaussian Elimination
    Solve the following linear systems:
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Solving a linear system with matrices using Gaussian elimination

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