SETASIGNThis e-book
Tutorial 2.31 Use a spreadsheet to solve systems of equations involving row operations
Suppose that we want to use a spreadsheet to solve:
3 4 10
2 2 15
3 4 2 10 x y z
x y z x y z
− + =
+ + =
− + =
by applying the Gauss-Jordan row operations. We would need to enter the numbers from the given problem…creating an augmented matrix (as can be seen in the partial screen shot).
Recall that the goal of the Gauss-Jordan row operations method is to convert all but the last column into an N×N identity matrix so that the solutions for the given system will appear in the last column.
For this example, we need a 3 × 3 identity matrix and the solutions will appear in the fourth column.
Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more
- © Photononstop
> Apply now
redefine your future
AxA globAl grAduAte progrAm 2015
axa_ad_grad_prog_170x115.indd 1 19/12/13 16:36
Recall that the goal of the first step in the Gauss-Jordan process is to get a 1 in the R1C1 position (cell D2 in the partial screen shot at the right) of the desired identity matrix. Since we already have that step accomplished, we then need to convert the other entries in C1 (cells D3 and D4) into zeroes:
• Click into cell D6 and type: =D2…and then copy this formula from cells D6 to G6.
• Click into cell D7 and type: =D3-2*D2…and then copy this formula from cells D7 to G7.
• Click into dell D8 and type: =D4–3*D2…and then copy this formula from cells D8 to G8.
We now have the first column of the desired identity matrix completed. The next step is to get a 1 into the R2C2 position (cell E7 in the partial screen shot above to the right). Since the non-zero entries of the third row are all multiplies of 5, it would be better to have these entries as the second row rather than the third. So, we can swap the last two rows around:
• Click into cell D10 and type: =D6…and then copy this formula from D10 to G10.
• Click into cell D11 and type: =D8…and then copy this formula from D11 to G11.
• Click into cell D12 and type: =D7…and then copy this formula from D12 to G12.
Now that we have the rows swapped around, we still need to get a 1 in the R2C2 position of the desired identity matrix. So, we need to divide the entire second row by 5:
• Click into cell D14 and type: =D10…and then copy this formula from D14 to G14.
• Click into cell D15 and type: =D11/5…and then copy this formula from D15 to G15.
• Click into cell D16 and type: =D12…and then copy this formula from D16 to G16.
171
We now have to convert the other entries in C2 (cells E14 and E16) into zeroes:
• Click into cell D18 and type: =D14+3*D15…and then copy this formula from D18 to G18.
• Click into cell D19 and type: =D15…and then copy this formula from D19 to G19
• Click into cell D20 and type: =D16–8*D15…and then copy this formula from D20 to G20.
We now have the first and the second columns of the desired identity matrix completed. The next step is to get a 1 into the R3C3 position of the desired identity matrix. Since there is a 9 in that position (cell F20 in the partial screen shot on the previous page), we just need to divide the third row by 9:
• Click into cell D22 and type: =D18…and then copy this formula from D22 to G22.
• Click into cell D23 and type: =D19…and then copy this formula from D23 to G23.
• Click into cell D24 and type: =D20/9…and then copy this formula from D24 to G24.
We now have to convert the other entries in C3 (cells F22 and F23) into zeroes:
• Click into cell D26 and type: =D22+2*D24…and then copy this formula from D26 to G26.
• Click into cell D27 and type: =D23+2*D24…and then copy this formula from D27 to G27.
• Click into cell D28 and type: =D24…and then copy this formula from D28 to G28.
We now have the desired identity matrix completely finished. The only thing to do is now read the solutions to the given system: x = 4 (based on the entry in cell G26)
y = 2 (based on the entry in cell G27) z = 3 (based on the entry in cell G28).
NOTE: This process can be duplicated using any sized system. The only difference will be the number of rows and columns that will be handled based on the size of the system.