Solving Linear Systems, Continued and The Inverse of a Matrix
Solving Linear
Systems
Math 240
Solving Linear
Systems
Gauss-Jordan
elimination
Rank
Inverse
matrices
Definition
Computing
inverses
Properties of
inverses
Using inverse
matrices
Conclusion
Solving Linear Systems, Continued
and
The Inverse of a Matrix
Math 240 ¡ª Calculus III
Summer 2015, Session II
Tuesday, July 7, 2015
Solving Linear
Systems
Agenda
Math 240
Solving Linear
Systems
Gauss-Jordan
elimination
Rank
Inverse
matrices
Definition
Computing
inverses
Properties of
inverses
Using inverse
matrices
Conclusion
1. Solving Linear Systems
Gauss-Jordan elimination
The rank of a matrix
2. The inverse of a square matrix
Definition
Computing inverses
Properties of inverses
Using inverse matrices
Conclusion
Solving Linear
Systems
Gaussian elimination
Math 240
Solving Linear
Systems
Gauss-Jordan
elimination
Rank
Inverse
matrices
Definition
Computing
inverses
Properties of
inverses
Using inverse
matrices
Conclusion
Gaussian elimination solves a linear system by reducing to
REF via elementary row ops and then using back substitution.
Example
3x1 ? 2x2 +
x1 ? 2x2 +
2x1 ? x2 ?
?
1
¡ú ?0
0
2x3 = 9
x3 = 5
2x3 = ?1
?2
1
0
1
3
1
?
?
3 ?2
2
9
?1 ?2
1
5?
2 ?1 ?2 ?1
?
5
x1 ? 2x2 + x3 = 5
5?
x2 + 3x3 = 5
2
x3 = 2
Steps
1. P12
3. A13 (?2)
5. A23 (?3)
2. A12 (?3)
4. A32 (?1)
6. M3 ?1
13
Back substitution gives the solution (1, ?1, 2).
Solving Linear
Systems
Gauss-Jordan elimination
Math 240
Solving Linear
Systems
Gauss-Jordan
elimination
Rank
Inverse
matrices
Definition
Computing
inverses
Properties of
inverses
Using inverse
matrices
Conclusion
Reducing the augmented matrix to RREF makes the system
even easier to solve.
Example
?
?
?
1 ?2 1 5
1
?0
1 3 5? ¡ú ? 0
0
0 1 2
0
0
1
0
?
0 1
0 ?1?
1 2
x1
x2
= 1
= ?1
x3 = 2
Steps
1. A32 (?3)
2. A31 (?1)
3. A21 (2)
Now, without any back substitution, we can see that the
solution is (1, ?1, 2).
The method of solving a linear system by reducing its
augmented matrix to RREF is called Gauss-Jordan
elimination.
Solving Linear
Systems
The rank of a matrix
Math 240
Solving Linear
Systems
Gauss-Jordan
elimination
Rank
Inverse
matrices
Definition
Computing
inverses
Properties of
inverses
Using inverse
matrices
Conclusion
Definition
The rank of a matrix, A, is the number of nonzero rows it has
after reduction to REF. It is denoted by rank(A).
If A is the coefficient matrix of an m ¡Á n linear system and
rank(A# ) = rank(A) = n then the REF looks like
?
?
1 ? ? ¡¤¡¤¡¤ ?
x1 = ?
? 1 ? . . . ??
?
?
x2 = ?
?
.. ?
..
?
?
..
.
.
?
?
.
?
1 ??
xn = ?
0 ........... 0
0
Lemma
Suppose Ax = b is an m ¡Á n linear system with augmented
matrix A# . If rank(A# ) = rank(A) = n then the system has a
unique solution.
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