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SYSTEMS OF LINEAR SIMULTANEOUS EQUATIONS
a11x1 +a12x2 . . . +a1nxn = b1a21x1 +a22x2 . . . +a2nxn = b2
......
......
...am1x1 +an2x2 . . . +amnxn = bm
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
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
SYSTEMS OF LINEAR SIMULTANEOUS EQUATIONS
a11x1 +a12x2 . . . +a1nxn = b1a21x1 +a22x2 . . . +a2nxn = b2
......
......
...am1x1 +an2x2 . . . +amnxn = bm
x =
x1x2...xn
SYSTEMS OF LINEAR SIMULTANEOUS EQUATIONS
a11x1 +a12x2 . . . +a1nxn = b1a21x1 +a22x2 . . . +a2nxn = b2
......
......
...am1x1 +an2x2 . . . +amnxn = bm
x =
x1x2...xn
SYSTEMS OF LINEAR SIMULTANEOUS EQUATIONS
a11x1 +a12x2 . . . +a1nxn = b1a21x1 +a22x2 . . . +a2nxn = b2
......
......
...am1x1 +an2x2 . . . +amnxn = bm
b =
b1b2...bn
SYSTEMS OF LINEAR SIMULTANEOUS EQUATIONS
a11x1 +a12x2 . . . +a1nxn = b1a21x1 +a22x2 . . . +a2nxn = b2
......
......
...am1x1 +an2x2 . . . +amnxn = bm
b =
b1b2...bn
SYSTEMS OF LINEAR SIMULTANEOUS EQUATIONS
a11x1 +a12x2 . . . +a1nxn = b1a21x1 +a22x2 . . . +a2nxn = b2
......
......
...am1x1 +an2x2 . . . +amnxn = bm
Ax = b
SYSTEMS OF LINEAR SIMULTANEOUS EQUATIONS
a11x1 +a12x2 . . . +a1nxn = b1a21x1 +a22x2 . . . +a2nxn = b2
......
......
...am1x1 +an2x2 . . . +amnxn = bm
Ax = b
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
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
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.
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.
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.
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.
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
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
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.
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.
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
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
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
