systems of equations
DESCRIPTION
Engineering MathematicsTRANSCRIPT
![Page 1: Systems of Equations](https://reader034.vdocument.in/reader034/viewer/2022051619/55cf98f4550346d0339aa9c8/html5/thumbnails/1.jpg)
SYSTEMS OF LINEAR SIMULTANEOUS EQUATIONS
a11x1 +a12x2 . . . +a1nxn = b1a21x1 +a22x2 . . . +a2nxn = b2
......
......
...am1x1 +an2x2 . . . +amnxn = bm
![Page 2: Systems of Equations](https://reader034.vdocument.in/reader034/viewer/2022051619/55cf98f4550346d0339aa9c8/html5/thumbnails/2.jpg)
SYSTEMS OF LINEAR SIMULTANEOUS EQUATIONS
a11x1 +a12x2 . . . +a1nxn = b1a21x1 +a22x2 . . . +a2nxn = b2
......
......
...am1x1 +an2x2 . . . +amnxn = bm
A =
a11 a12 a13 . . . a1na21 a22 a23 . . . a2n
......
.... . .
...am1 am2 am3 . . . amn
![Page 3: Systems of Equations](https://reader034.vdocument.in/reader034/viewer/2022051619/55cf98f4550346d0339aa9c8/html5/thumbnails/3.jpg)
SYSTEMS OF LINEAR SIMULTANEOUS EQUATIONS
a11x1 +a12x2 . . . +a1nxn = b1a21x1 +a22x2 . . . +a2nxn = b2
......
......
...am1x1 +an2x2 . . . +amnxn = bm
A =
a11 a12 a13 . . . a1na21 a22 a23 . . . a2n
......
.... . .
...am1 am2 am3 . . . amn
![Page 4: Systems of Equations](https://reader034.vdocument.in/reader034/viewer/2022051619/55cf98f4550346d0339aa9c8/html5/thumbnails/4.jpg)
SYSTEMS OF LINEAR SIMULTANEOUS EQUATIONS
a11x1 +a12x2 . . . +a1nxn = b1a21x1 +a22x2 . . . +a2nxn = b2
......
......
...am1x1 +an2x2 . . . +amnxn = bm
x =
x1x2...xn
![Page 5: Systems of Equations](https://reader034.vdocument.in/reader034/viewer/2022051619/55cf98f4550346d0339aa9c8/html5/thumbnails/5.jpg)
SYSTEMS OF LINEAR SIMULTANEOUS EQUATIONS
a11x1 +a12x2 . . . +a1nxn = b1a21x1 +a22x2 . . . +a2nxn = b2
......
......
...am1x1 +an2x2 . . . +amnxn = bm
x =
x1x2...xn
![Page 6: Systems of Equations](https://reader034.vdocument.in/reader034/viewer/2022051619/55cf98f4550346d0339aa9c8/html5/thumbnails/6.jpg)
SYSTEMS OF LINEAR SIMULTANEOUS EQUATIONS
a11x1 +a12x2 . . . +a1nxn = b1a21x1 +a22x2 . . . +a2nxn = b2
......
......
...am1x1 +an2x2 . . . +amnxn = bm
b =
b1b2...bn
![Page 7: Systems of Equations](https://reader034.vdocument.in/reader034/viewer/2022051619/55cf98f4550346d0339aa9c8/html5/thumbnails/7.jpg)
SYSTEMS OF LINEAR SIMULTANEOUS EQUATIONS
a11x1 +a12x2 . . . +a1nxn = b1a21x1 +a22x2 . . . +a2nxn = b2
......
......
...am1x1 +an2x2 . . . +amnxn = bm
b =
b1b2...bn
![Page 8: Systems of Equations](https://reader034.vdocument.in/reader034/viewer/2022051619/55cf98f4550346d0339aa9c8/html5/thumbnails/8.jpg)
SYSTEMS OF LINEAR SIMULTANEOUS EQUATIONS
a11x1 +a12x2 . . . +a1nxn = b1a21x1 +a22x2 . . . +a2nxn = b2
......
......
...am1x1 +an2x2 . . . +amnxn = bm
Ax = b
![Page 9: Systems of Equations](https://reader034.vdocument.in/reader034/viewer/2022051619/55cf98f4550346d0339aa9c8/html5/thumbnails/9.jpg)
SYSTEMS OF LINEAR SIMULTANEOUS EQUATIONS
a11x1 +a12x2 . . . +a1nxn = b1a21x1 +a22x2 . . . +a2nxn = b2
......
......
...am1x1 +an2x2 . . . +amnxn = bm
Ax = b
![Page 10: Systems of Equations](https://reader034.vdocument.in/reader034/viewer/2022051619/55cf98f4550346d0339aa9c8/html5/thumbnails/10.jpg)
SYSTEMS OF LINEAR SIMULTANEOUS EQUATIONS
a11x1 +a12x2 . . . +a1nxn = b1a21x1 +a22x2 . . . +a2nxn = b2
......
......
...am1x1 +an2x2 . . . +amnxn = bm
I One solution
I No solution
I Infinitely many solutions
![Page 11: Systems of Equations](https://reader034.vdocument.in/reader034/viewer/2022051619/55cf98f4550346d0339aa9c8/html5/thumbnails/11.jpg)
SYSTEMS OF LINEAR SIMULTANEOUS EQUATIONS
a11x1 +a12x2 . . . +a1nxn = b1a21x1 +a22x2 . . . +a2nxn = b2
......
......
...am1x1 +an2x2 . . . +amnxn = bm
I One solution
I No solution
I Infinitely many solutions
![Page 12: Systems of Equations](https://reader034.vdocument.in/reader034/viewer/2022051619/55cf98f4550346d0339aa9c8/html5/thumbnails/12.jpg)
SYSTEMS OF LINEAR SIMULTANEOUS EQUATIONS
a11x1 +a12x2 . . . +a1nxn = b1a21x1 +a22x2 . . . +a2nxn = b2
......
......
...an1x1 +an2x2 . . . +annxn = bn
Ax = b
A−1Ax = A−1b
x = A−1b
One solution when A is invertible. The other two scenarios occurwhen A is not invertible.
![Page 13: Systems of Equations](https://reader034.vdocument.in/reader034/viewer/2022051619/55cf98f4550346d0339aa9c8/html5/thumbnails/13.jpg)
SYSTEMS OF LINEAR SIMULTANEOUS EQUATIONS
a11x1 +a12x2 . . . +a1nxn = b1a21x1 +a22x2 . . . +a2nxn = b2
......
......
...an1x1 +an2x2 . . . +annxn = bn
Ax = b
A−1Ax = A−1b
x = A−1b
One solution when A is invertible. The other two scenarios occurwhen A is not invertible.
![Page 14: Systems of Equations](https://reader034.vdocument.in/reader034/viewer/2022051619/55cf98f4550346d0339aa9c8/html5/thumbnails/14.jpg)
SYSTEMS OF LINEAR SIMULTANEOUS EQUATIONS
a11x1 +a12x2 . . . +a1nxn = b1a21x1 +a22x2 . . . +a2nxn = b2
......
......
...an1x1 +an2x2 . . . +annxn = bn
Ax = b
A−1Ax = A−1b
x = A−1b
One solution when A is invertible. The other two scenarios occurwhen A is not invertible.
![Page 15: Systems of Equations](https://reader034.vdocument.in/reader034/viewer/2022051619/55cf98f4550346d0339aa9c8/html5/thumbnails/15.jpg)
SYSTEMS OF LINEAR SIMULTANEOUS EQUATIONS
a11x1 +a12x2 . . . +a1nxn = b1a21x1 +a22x2 . . . +a2nxn = b2
......
......
...an1x1 +an2x2 . . . +annxn = bn
Ax = b
A−1Ax = A−1b
x = A−1b
One solution when A is invertible. The other two scenarios occurwhen A is not invertible.
![Page 16: Systems of Equations](https://reader034.vdocument.in/reader034/viewer/2022051619/55cf98f4550346d0339aa9c8/html5/thumbnails/16.jpg)
SYSTEMS OF LINEAR SIMULTANEOUS EQUATIONS -Elementary Row Operations
a11x1 +a12x2 . . . +a1nxn = b1a21x1 +a22x2 . . . +a2nxn = b2
......
...... =
...am1x1 +an2x2 . . . +amnxn = bm
1. Augmented matrix (A|b).
a11 a12 a13 . . . a1n b1a21 a22 a23 . . . a2n b2
......
.... . .
......
am1 am2 am3 . . . amn bm
![Page 17: Systems of Equations](https://reader034.vdocument.in/reader034/viewer/2022051619/55cf98f4550346d0339aa9c8/html5/thumbnails/17.jpg)
SYSTEMS OF LINEAR SIMULTANEOUS EQUATIONS -Elementary Row Operations
a11x1 +a12x2 . . . +a1nxn = b1a21x1 +a22x2 . . . +a2nxn = b2
......
...... =
...am1x1 +an2x2 . . . +amnxn = bm
1. Augmented matrix (A|b).
a11 a12 a13 . . . a1n b1a21 a22 a23 . . . a2n b2
......
.... . .
......
am1 am2 am3 . . . amn bm
![Page 18: Systems of Equations](https://reader034.vdocument.in/reader034/viewer/2022051619/55cf98f4550346d0339aa9c8/html5/thumbnails/18.jpg)
SYSTEMS OF LINEAR SIMULTANEOUS EQUATIONS
a11x1 +a12x2 . . . +a1nxn = b1a21x1 +a22x2 . . . +a2nxn = b2
......
...... =
...am1x1 +an2x2 . . . +amnxn = bm
1. Augmented matrix (A|b).
2. Apply Elementary Row Operations to reduce it to one inwhich A is upper triangular.
![Page 19: Systems of Equations](https://reader034.vdocument.in/reader034/viewer/2022051619/55cf98f4550346d0339aa9c8/html5/thumbnails/19.jpg)
Elementary Row Operations
ERO1 Interchange two rows.
ERO2 Multiply a row by a non-zero number.
ERO3 Add a multiple of one row to another.
![Page 20: Systems of Equations](https://reader034.vdocument.in/reader034/viewer/2022051619/55cf98f4550346d0339aa9c8/html5/thumbnails/20.jpg)
Elementary Row Operations
a11 a12 a13 . . . a1n b1a21 a22 a23 . . . a2n b2
......
.... . .
......
am1 am2 am3 . . . amn bm
a′11 a12′ a′13 . . . a′1n b′10 a′22 a′23 . . . a′2n b′2...
......
. . ....
...0 0 0 . . . a′mn b′m
![Page 21: Systems of Equations](https://reader034.vdocument.in/reader034/viewer/2022051619/55cf98f4550346d0339aa9c8/html5/thumbnails/21.jpg)
Elementary Row Operations
a11 a12 a13 . . . a1n b1a21 a22 a23 . . . a2n b2
......
.... . .
......
am1 am2 am3 . . . amn bm
a′11 a12′ a′13 . . . a′1n b′10 a′22 a′23 . . . a′2n b′2...
......
. . ....
...0 0 0 . . . a′mn b′m
![Page 22: Systems of Equations](https://reader034.vdocument.in/reader034/viewer/2022051619/55cf98f4550346d0339aa9c8/html5/thumbnails/22.jpg)
SYSTEMS OF LINEAR SIMULTANEOUS EQUATIONS
a11x1 +a12x2 . . . +a1nxn = b1a21x1 +a22x2 . . . +a2nxn = b2
......
...... =
...am1x1 +an2x2 . . . +amnxn = bm
1. Augmented matrix (A|b).
2. Apply Elementary Row Operations to reduce it to one inwhich A is upper triangular.
3. Back-substitution.
![Page 23: Systems of Equations](https://reader034.vdocument.in/reader034/viewer/2022051619/55cf98f4550346d0339aa9c8/html5/thumbnails/23.jpg)
SYSTEMS OF LINEAR SIMULTANEOUS EQUATIONS
Row reduced coefficient matrix:a′11 a12′ a′13 . . . a′1n b′10 a′22 a′23 . . . a′2n b′2...
......
. . ....
...0 0 0 . . . a′mn b′m
0x1 + 0x2 + . . . + 0xn−1 + a′mnxn = b′m
If a′mn 6= 0, then one solution.
![Page 24: Systems of Equations](https://reader034.vdocument.in/reader034/viewer/2022051619/55cf98f4550346d0339aa9c8/html5/thumbnails/24.jpg)
SYSTEMS OF LINEAR SIMULTANEOUS EQUATIONS
Row reduced coefficient matrix:a′11 a12′ a′13 . . . a′1n b′10 a′22 a′23 . . . a′2n b′2...
......
. . ....
...0 0 0 . . . a′mn b′m
0x1 + 0x2 + . . . + 0xn−1 + a′mnxn = b′m
If a′mn 6= 0, then one solution.
![Page 25: Systems of Equations](https://reader034.vdocument.in/reader034/viewer/2022051619/55cf98f4550346d0339aa9c8/html5/thumbnails/25.jpg)
SYSTEMS OF LINEAR SIMULTANEOUS EQUATIONS
Row reduced coefficient matrix:a′11 a12′ a′13 . . . a′1n b′10 a′22 a′23 . . . a′2n b′2...
......
. . ....
...0 0 0 . . . a′mn b′m
0x1 + 0x2 + . . . + 0xn−1 + a′mnxn = b′m
If a′mn 6= 0, then one solution.
![Page 26: Systems of Equations](https://reader034.vdocument.in/reader034/viewer/2022051619/55cf98f4550346d0339aa9c8/html5/thumbnails/26.jpg)
SYSTEMS OF LINEAR SIMULTANEOUS EQUATIONSRow reduced coefficient matrix:
a′11 a12′ a′13 . . . a′1n b′10 a′22 a′23 . . . a′2n b′2...
......
. . ....
...0 0 0 . . . a′mn b′m
0x1 + 0x2 + . . . + 0xn−1 + a′mnxn = b′m
If a′mn = 0, then0 · xn = b′m.
Case 1.: b′m 6= 0, then no solution, equations are inconsistent.Case 2.: b′m = 0, then infinite number of solution, equations areconsistent.
![Page 27: Systems of Equations](https://reader034.vdocument.in/reader034/viewer/2022051619/55cf98f4550346d0339aa9c8/html5/thumbnails/27.jpg)
SYSTEMS OF LINEAR SIMULTANEOUS EQUATIONSRow reduced coefficient matrix:
a′11 a12′ a′13 . . . a′1n b′10 a′22 a′23 . . . a′2n b′2...
......
. . ....
...0 0 0 . . . a′mn b′m
0x1 + 0x2 + . . . + 0xn−1 + a′mnxn = b′m
If a′mn = 0, then0 · xn = b′m.
Case 1.: b′m 6= 0, then no solution, equations are inconsistent.Case 2.: b′m = 0, then infinite number of solution, equations areconsistent.
![Page 28: Systems of Equations](https://reader034.vdocument.in/reader034/viewer/2022051619/55cf98f4550346d0339aa9c8/html5/thumbnails/28.jpg)
SYSTEMS OF LINEAR SIMULTANEOUS EQUATIONSRow reduced coefficient matrix:
a′11 a12′ a′13 . . . a′1n b′10 a′22 a′23 . . . a′2n b′2...
......
. . ....
...0 0 0 . . . a′mn b′m
0x1 + 0x2 + . . . + 0xn−1 + a′mnxn = b′m
If a′mn = 0, then0 · xn = b′m.
Case 1.: b′m 6= 0, then no solution, equations are inconsistent.Case 2.: b′m = 0, then infinite number of solution, equations areconsistent.
![Page 29: Systems of Equations](https://reader034.vdocument.in/reader034/viewer/2022051619/55cf98f4550346d0339aa9c8/html5/thumbnails/29.jpg)
SYSTEMS OF LINEAR SIMULTANEOUS EQUATIONSRow reduced coefficient matrix:
a′11 a12′ a′13 . . . a′1n b′10 a′22 a′23 . . . a′2n b′2...
......
. . ....
...0 0 0 . . . a′mn b′m
0x1 + 0x2 + . . . + 0xn−1 + a′mnxn = b′m
If a′mn = 0, then0 · xn = b′m.
Case 1.: b′m 6= 0, then no solution, equations are inconsistent.
Case 2.: b′m = 0, then infinite number of solution, equations areconsistent.
![Page 30: Systems of Equations](https://reader034.vdocument.in/reader034/viewer/2022051619/55cf98f4550346d0339aa9c8/html5/thumbnails/30.jpg)
SYSTEMS OF LINEAR SIMULTANEOUS EQUATIONSRow reduced coefficient matrix:
a′11 a12′ a′13 . . . a′1n b′10 a′22 a′23 . . . a′2n b′2...
......
. . ....
...0 0 0 . . . a′mn b′m
0x1 + 0x2 + . . . + 0xn−1 + a′mnxn = b′m
If a′mn = 0, then0 · xn = b′m.
Case 1.: b′m 6= 0, then no solution, equations are inconsistent.Case 2.: b′m = 0, then infinite number of solution, equations areconsistent.
![Page 31: Systems of Equations](https://reader034.vdocument.in/reader034/viewer/2022051619/55cf98f4550346d0339aa9c8/html5/thumbnails/31.jpg)
SYSTEMS OF LINEAR SIMULTANEOUS EQUATIONS -Rank
Row reduced coefficient matrix:a′11 a12′ a′13 . . . a′1n b′10 a′22 a′23 . . . a′2n b′2...
......
. . ....
...0 0 0 . . . a′mn b′m
Rank of matrix A is the number of non-zero rows in the rowreduced coefficient matrix at the end of the forward reduction.Note: If A is and n × n invertible matrix, then its rank is n.
![Page 32: Systems of Equations](https://reader034.vdocument.in/reader034/viewer/2022051619/55cf98f4550346d0339aa9c8/html5/thumbnails/32.jpg)
SYSTEMS OF LINEAR SIMULTANEOUS EQUATIONS
a11x1 +a12x2 . . . +a1nxn = 0a21x1 +a22x2 . . . +a2nxn = 0
......
......
...an1x1 +an2x2 . . . +annxn = 0
Ax = 0
Note:
1. x = 0 is always a solution.
2. If A is square and invertible, then x = 0 is the only solution.
3. If A is not, then there may be other solutions.
![Page 33: Systems of Equations](https://reader034.vdocument.in/reader034/viewer/2022051619/55cf98f4550346d0339aa9c8/html5/thumbnails/33.jpg)
SYSTEMS OF LINEAR SIMULTANEOUS EQUATIONS
a11x1 +a12x2 . . . +a1nxn = 0a21x1 +a22x2 . . . +a2nxn = 0
......
......
...an1x1 +an2x2 . . . +annxn = 0
Ax = 0
Note:
1. x = 0 is always a solution.
2. If A is square and invertible, then x = 0 is the only solution.
3. If A is not, then there may be other solutions.
![Page 34: Systems of Equations](https://reader034.vdocument.in/reader034/viewer/2022051619/55cf98f4550346d0339aa9c8/html5/thumbnails/34.jpg)
SYSTEMS OF LINEAR SIMULTANEOUS EQUATIONS
a11x1 +a12x2 . . . +a1nxn = 0a21x1 +a22x2 . . . +a2nxn = 0
......
......
...an1x1 +an2x2 . . . +annxn = 0
Ax = 0
Note:
1. x = 0 is always a solution.
2. If A is square and invertible, then x = 0 is the only solution.
3. If A is not, then there may be other solutions.
![Page 35: Systems of Equations](https://reader034.vdocument.in/reader034/viewer/2022051619/55cf98f4550346d0339aa9c8/html5/thumbnails/35.jpg)
SYSTEMS OF LINEAR SIMULTANEOUS EQUATIONS
a11x1 +a12x2 . . . +a1nxn = 0a21x1 +a22x2 . . . +a2nxn = 0
......
......
...an1x1 +an2x2 . . . +annxn = 0
Ax = 0
Note:
1. x = 0 is always a solution.
2. If A is square and invertible, then x = 0 is the only solution.
3. If A is not, then there may be other solutions.
![Page 36: Systems of Equations](https://reader034.vdocument.in/reader034/viewer/2022051619/55cf98f4550346d0339aa9c8/html5/thumbnails/36.jpg)
SYSTEMS OF LINEAR SIMULTANEOUS EQUATIONS
a11x1 +a12x2 . . . +a1nxn = 0a21x1 +a22x2 . . . +a2nxn = 0
......
......
...an1x1 +an2x2 . . . +annxn = 0
Ax = 0
Note:
1. x = 0 is always a solution.
2. If A is square and invertible, then x = 0 is the only solution.
3. If A is not, then there may be other solutions.