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System of Linear Equations & Gauss Elimination Method
3 Variables SLEs in Matrix Form
a11x + a12y + a13z = b1a21x + a22y + a23z = b2a31x + a32y + a33z = b3
𝑎11 𝑎12 𝑎13𝑎21 𝑎22 𝑎23𝑎31 𝑎32 𝑎33
𝑥𝑦𝑧
=𝑏1𝑏2𝑏3
A X = B
𝐴/𝐵 =𝑎11𝑎21𝑎31
𝑎12𝑎22𝑎32
𝑎13𝑎23𝑎33
𝑏1𝑏2𝑏3
Augmented Matrix,
R1
R2
R3
R1 R1/a11, 𝐴/𝐵 ≅1𝑎21𝑎31
𝑎12′𝑎22𝑎32
𝑎13′𝑎23𝑎33
𝑏1′𝑏2𝑏3
Gauss Elimination Method,
R2 R2 – a21R1, 𝐴/𝐵 ≅10𝑎31
𝑎12′𝑎22′𝑎32
𝑎13′𝑎23′𝑎33
𝑏1′𝑏2′𝑏3
R3 R3 – a31R1, 𝐴/𝐵 ≅100
𝑎12′𝑎22′𝑎32′
𝑎13′𝑎23′𝑎33′
𝑏1′𝑏2′𝑏3′
R2 R2/a22’, 𝐴/𝐵 ≅100
𝑎12′1
𝑎32′
𝑎13′𝑎23′′𝑎33′
𝑏1′𝑏2′′𝑏3′
R3 R3 – a32’R2, 𝐴/𝐵 ≅100
𝑎12′10
𝑎13′𝑎23′′𝑎33′′
𝑏1′𝑏2′′𝑏3′′
Case 1
𝐴/𝐵 ≅100
𝑎12′10
𝑎13′𝑎23′′𝑎33′′
𝑏1′𝑏2′′𝑏3′′
r(A) = r(A/B) = 3
Case 2 r(A) = 2 & r(A/B) = 3 r(A) ≠ r(A/B)
Case 3 r(A) = r(A/B) = 2
a33’’ ≠ 0 Unique Solution
𝐴 ≅100
𝑎12′10
𝑎13′
𝑎23′′
𝑎33′′
a33’’ = 0 b3’’ ≠ 0 No Solution
a33’’ = 0 b3’’ = 0 Infinitely Many Solutions
1 𝑎12′ 𝑎13′0 1 𝑎23′′0 0 𝑎33′′
𝑥𝑦𝑧
=𝑏1′𝑏2′′𝑏3′′
a33’’z = b3’’ z = b3’’/a33’’= L
y + a23’’z = b2’’y = b2’’ - a23’’z = b2’’ - a23’’L = M
x + a12’y + a13’z = b1’ x = b1’ - a12’y - a13’z = b1’ - a12’M - a13’L = N
Consistency of SLEs (Non Homogeneous)
• SLE is Consistent if r(A) = r(A/B)
• If r(A) = r(A/B) = n (number of variables) then SLE has unique solution
• If r(A) = r(A/B) < n then SLE has infinitely many solutions
• SLE is inconsistent (no solution) if r(A) ≠ r(A/B)
Example 1 Test the Consistency and Solve the following SLEs using Gauss Elimination Method if possible:
x + y + z = 6x - y + 2z = 53x + y + z = 8
𝐴/𝐵 =113
1−11
121
658
Augmented Matrix,
1 1 11 −1 23 1 1
𝑥𝑦𝑧
=658
In Matrix form SLEs is, A X = B
R2 R2 – R1, 𝐴/𝐵 ≅103
1−21
111
6−18
R3 R3 – 3R1, 𝐴/𝐵 ≅100
1−2−2
11−2
6−1−10
R3 R3 – R2, 𝐴/𝐵 ≅100
1−20
11−3
6−1−9
R2 R2/(-2), 𝐴/𝐵 ≅100
110
1−1/21
61/23R3 R3/(-3),
Therefore, Solution of the system is unique and it is (x, y , z) = (1, 2, 3).
r(A) = r(A/B) = 3 = n, Therefore, System is Consistent & It has Unique Solution.
1 1 10 1 −1/20 0 1
𝑥𝑦𝑧
=61/23
z = 3
y -1/2 z = 1/2 y = 1/2 + (1/2) (z) = 1/2 + (1/2) (3) = (3+1)/2 = 4/2 = 2
x + y + z = 6 x = 6 - y - z = 6 – 2 – 3 = 1
Example 2 Test the Consistency and Solve the following SLEs using Gauss Elimination Method if possible:
3x + 2y - 5z = 4x + y - 2z = 15x + 3y - 8z = 6
𝐴/𝐵 =315
213
−5−2−8
416
Augmented Matrix,
3 2 −51 1 −25 3 −8
𝑥𝑦𝑧
=416
In Matrix form SLEs is, A X = B
R1 R2, 𝐴/𝐵 ≅135
123
−2−5−8
146
R2 R2 – 3R1, 𝐴/𝐵 ≅105
1−13
−21−8
116
R3 R3 – 5R1, 𝐴/𝐵 ≅100
1−1−2
−212
111
R2 R2/(-1), 𝐴/𝐵 ≅100
11−2
−2−12
1−11
r(A) = 2
Therefore, System is inconsistent & It has no solution.
1 1 −20 1 −10 0 0
𝑥𝑦𝑧
=1−1−1
R3 R3 + 2R2, 𝐴/𝐵 ≅100
110
−2−10
1−1−1
by row (3), 0 z = -10 = -1 which Is not possible
Therefor, Solution of given SLE is not possible.
r(A/B) = 3 ∴r(A) ≠ r(A/B)&
Example 3 Test the Consistency and Solve the following SLEs using Gauss Elimination Method if possible:
2x + 2y + 2z = 0-2x + 5y + 2z = 18x + y + 4z = -1
𝐴/𝐵 =2−28
251
224
01−1
Augmented Matrix,
2 2 2−2 5 28 1 4
𝑥𝑦𝑧
=01−1
In Matrix form SLEs is, A X = B
R1 R1/2,
R2 R2 + 2R1,
R3 R3 – 8R1,
R3 R3 + R2,
𝐴/𝐵 ≅1−28
151
124
01−1
𝐴/𝐵 ≅108
171
144
01−1
𝐴/𝐵 ≅100
17−7
14−4
01−1
𝐴/𝐵 ≅100
170
140
010
∴(x, y , z) = {(−𝟏
𝟕−
𝟑
𝟕𝒌,
𝟏
𝟕−
𝟒
𝟕𝒌, k) / k є R}.
r(A) = r(A/B) = 2 < n,
Therefore, System is Consistent & It has Infinitely Many Solutions.1 1 10 1 4/70 0 0
𝑥𝑦𝑧
=01/70
z = k, k є R y + 4/7 z = 1/7
y = (1/7) - (4/7) (z) = (1/7) - (4/7) (k) = 𝟏
𝟕−
𝟒
𝟕𝒌, k є R
x + y + z = 0 x = 0 - y - z = 0 – (𝟏
𝟕−
𝟒
𝟕𝒌 ) – k = −
𝟏
𝟕+
𝟒
𝟕𝒌 − 𝒌 = −
𝟏
𝟕−
𝟑
𝟕𝒌
R3 R2/7, 𝐴/𝐵 ≅100
110
14/70
01/70
Example 4 Test the Consistency and Solve the following SLEs using Gauss Elimination Method if possible:
3x + y = -5-6x - 2y = 104x + 5y = 8
𝐴/𝐵 =3 1 −5−6 −2 104 5 8
Augmented Matrix,
3 1−6 −24 5
𝑥𝑦 =
−5108
In Matrix form SLEs is, A X = B
R1 R1/3,
R2 R2 + 6R1,
R3 R3 – 4R1,
R2 R3,
𝐴/𝐵 ≅1 1/3 −5/3−6 −2 104 5 8
𝐴/𝐵 ≅1 1/3 −5/30 0 04 5 8
𝐴/𝐵 ≅1 1/3 −5/30 0 00 11/3 44/3
𝐴/𝐵 ≅1 1/3 −5/30 11/3 44/30 0 0
Therefore, Solution of the system is unique and it is (x, y) = (-3, 4).
r(A) = r(A/B) = 2 = n, Therefore, System is Consistent & It has Unique Solution.
y = 4
x + y/3 = -5/3
R2 R2/(11/3), 𝐴/𝐵 ≅1 1/3 −5/30 1 40 0 0
1 1/30 10 0
𝑥𝑦 =
−5/340
x = -5/3 - y/3 x = -5/3 - 4/3 x = -9/3 = -3
Example 5 Test the Consistency and Solve the following SLEs using Gauss Elimination Method if possible:
x - 2y + w = 3-x + 2y + z – ½ w = -74x - 8y + 6z + 7w = -3
𝐴/𝐵 =1−14
−22−8
016
1 3−1/2 −7
7 −3
Augmented Matrix,
1−14
−22−8
016
1−1/27
𝑥𝑦𝑧𝑤
=3−7−3
In Matrix form SLEs is, A X = B
R2 R2 + R1,
R3 R3 – 4R1,
𝐴/𝐵 ≅104
−20−8
016
1 31/2 −4
7 −3
𝐴/𝐵 ≅100
−200
016
1 31/2 −4
3 −15
R3 R3 – 6R2, 𝐴/𝐵 ≅100
−200
010
1 31/2 −4
0 9
r(A) = 2 r(A/B) = 3 ∴r(A) ≠ r(A/B)&
Therefore, System is inconsistent & It has no solution.
Example 6 Test the Consistency and Solve the following SLEs using Gauss Elimination Method if possible:
3x + 2y + w = 162x + y + 3z = 162x + 12z -5w = 5
Augmented Matrix,
In Matrix form SLEs is, A X = B322
210
0312
10−5
𝑥𝑦𝑧𝑤
=16165
𝐴/𝐵 =322
210
0312
1 160 16−5 5
R1 R1 - R2,
R2 R2 - 2R1,
R3 R3 – 2R1,
R2 R2/(-1),
𝐴/𝐵 ≅122
110
−3312
1 00 16−5 5
𝐴/𝐵 ≅102
1−10
−3912
1 0−2 16−5 5
𝐴/𝐵 ≅100
1−1−2
−3918
1 0−2 16−7 5
𝐴/𝐵 ≅100
11−2
−3−918
1 02 −16−7 5
∴(x, y , z, w) = {(−𝟔𝒌 + 𝟐𝟓, 𝟗𝒌 − 𝟑𝟒, k, 9) / k є R}.
r(A) = r(A/B) = 3 < 4 = n,
Therefore, System is Consistent & It has Infinitely Many Solutions.
z = k, k є R
y - 9z + 2w = -16 y = (-16) + (9)(z) – 2w = (-16) + (9)(k) – 18 = 𝟗𝒌 − 𝟑𝟒, k є R
x + y - 3z + w = 0 x = 0 - y + 3z – w = 0 – (9k-34) + 3k – 9 = −𝟔𝒌 + 𝟐𝟓, 𝐤є R
R3 R3 + 2R2, 𝐴/𝐵 ≅100
110
−3−90
1 02 −16−3 −27
100
110
−3−90
121
𝑥𝑦𝑧𝑤
=0
−169
& w = 9
Next Lecture : Homogeneous SLES