Solve the following system of simultaneous equations by the Gauss-Jordan method. 5 x 1 − 2 x 2 + 6 x 3 = 0 −2 x 1 + 4 x 2 + x 3 + 3 x 4 = 18 6 x 1 + x 2 + 6 x 3 + 8 x 4 = −29 3 x 2 + 8 x 3 + 7 x 4 = 11

Use the Gauss Jordan method to solve the simultaneous equations.

Given information:

   5x1−2x2+6x3             =0−2x1+4x2+   x3+3x4=18   6x1+    x2+6x3+8x4=−29               3x2+8x3+7x4=11

Calculation:

Write the system in the matrix form:

Ax=P[5−260−241361680387][x1x2x3x4]=[018−2911]

Write the augmented matrix:

[5−260−241361680387|018−2911]

Divide the first row by 5.

[1−25650−241361680387|018−2911]

In the second row, subtract the first row multiplied with ­2 from second row.

[1−25650−241361680387|018−2911]R2→R2−(−2)R1[1−25650−2−(−2×1)4−(−2×−25)1−(−2×65)361680387|018−2911][1−256500165175361680387|018−2911]

In the third row, subtract the first row multiplied with ­6 from third row.

[1−256500165175361680387|018−2911]R3→R3−6R1[1−25650016517536−6×11−6×(−25)6−6×(65)8−6×(0)0387|018−2911][1−25650016517530175−6580387|018−2911]

Divide the second row by 16/5.

[1−2565001171615160175−6580387|0458−2911]

In the first row, subtract the second row multiplied with −25 from first row.

[1−2565001171615160175−6580387|0458−2911]R1→R1−(−25R2)[1−25−(−25×1)65−(−25×1716)0−(−25×1516)01171615160175−6580387|0−(−25×458)458−2911][101383801171615160175−6580387|94458−2911]

In the third row, subtract the second row multiplied with 175  from third row.

[101383801171615160175−6580387|94458−2911]R3→R3−(175)R2[101383801171615160175−(175×1)−65−(175×1716)8−(175×1516)0387|94458−29−(175×458)11][1013838011716151600−771677160387|94458−385811]

In the fourth row, subtract the second row multiplied with 3 from fourth row.

[1013838011716151600−771677160387|94458−385811]R4→R4−3R2[1013838011716151600−7716771603−3×18−3×17167−3×1516|94458−385811−3×458][1013838011716151600−771677160077166716|94458−3858−478]

Divide the third row by −7716