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# Matrices : Gauss-Jordan Elimination

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65. A father, when dying, gave to his sons 30 barrels, of which 10 were full of wine, 10 were half full, and the last 10 were empty. Divide the wine and flasks so that there will be equal division among the three sons of both wine and barrels. Find all the solutions of the problem. (from Alcuin)

4, 5 Find all solutions of the equations using Gauss-Jordan elimination.

4. &#9474; x + y = 1 &#9474;
&#9474;2x - y = 5 &#9474;
&#9474;3x + 4y = 2&#9474;

5. &#9474; x3 + x4 = 0&#9474;
&#9474; x2 + x3 = 0&#9474;
&#9474;x1 + x2 = 0&#9474;
&#9474;x1 + x4 = 0&#9474;

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4-
First we define the augmented matrix as follow:

Now we do some elementary operations (R shows the rows):
1- -2R1+R2 ïƒ  R2:

2- -3R1+R3 ïƒ  R3:

3- 1/3R2+R3 ïƒ  R3:

4- -1/3R2 ïƒ  R2:

5- -R2+R1 ïƒ  R1:

We can see that the above matrix is in "Row Reduced Echolen Form" (RREF). Now we have the solution to this system. That is x=2 and y=-1.

5-
Again we must write the matrix in its augmented from, but note that the order of equations in the system does not matter at all, so we consider the last equation first and the one before last second and so on:

Now we do some elementary operations:
1- -R1+R2 ïƒ  R2:

2- -R2+R3 ïƒ  R3:

3- -R3+R4 ïƒ  R4:

At this point the ...

#### Solution Summary

Gauss-Jordan elimination is performed on matrices. A seven equation, nine unknown word problem is solved.

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