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**Unformatted text preview: **University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT A23 Winter 2008
Lecture 7
Sophie Chrysostomou
1.4 Solving Systems of Linear Equations The Connection Between Linear Systems of Equations and Matrices
Suppose that a11 a12 · · · a1n a21 a22 · · · a2n A= .
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am1 am2 · · · amn , x= x1
x2
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. ,b = b1
b2
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. bm xn what is the meaning of Ax = b? definition 0.1. Suppose we have the linear system
a11 x1 + a12 x2 + ... + a1n xn
a21 x1 + a22 x2 + ... + a2n xn
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. =
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. b1
b2
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. am1 x1 + am2 x2 + ... + amn xn = bm
This is equivalent to Ax = b a11 a12 · · · a21 a22 · · · A= .
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am1 am2 · · · where a1 n
a2 n . ,
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amn
1 x= x1
x2
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xn ,b = b1
b2
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bm A is called the coeﬃcient matrix a11 a21 A= .
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am1 of the system and
· · · a1 n
· · · a2 n
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· · · amn a12
a22
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am2 b1
b2
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. bm is called the augmented or partitioned matrix of the system.
We may solve the linear system using the operations:
1. Switch two equations.
2. Multiply an equation by a nonzero constant.
3. Replace an equation by that equation plus a multiple of another equation.
OR we can use the augmented matrix and use the following operations : Elementary Row Operations
a) (Row Interchange) Interchange the ith and j th row
(Ri ←→ Rj )
vectors of the matrix.
b) (Row scaling) Multiply the ith row of the matrix with a (Ri → aRi )
nonzero scalar a
c) (Row addition) Add to the ith row of the matrix s times (Ri → R1 + sRj )
the j th row. definition 0.2. If a matrix B can be obtained from A by performing a
sequence of elementary operations, then A and B are called row equivalent.
theorem 0.3. If [A|b] and [H |a] are row equivalent matrices then Ax = b
and H x = a have the same solution set. 2 example 0.4. Find the solution to the system
2x1 + 3x2 = 9
x1 − 2x2 = 1 definition 0.5. Row Echelon Form, Pivot. A matrix is in a row echelon
form (REF) if:
a) All rows containing only zeros are below rows with nonzero entries.
b) The ﬁrst nonzero entry in any row appears in a column to the right of the
ﬁrst nonzero entry in any row above it.
If a matrix is in a row echelon form, then the ﬁrst nonzero entry in a row is
called the pivot of that row.
example 0.6. 3 definition 0.7. Reduced Row Echelon Form, Pivot. A matrix is in a reduced row echelon form (RREF) if:
a) The matrix is in a row echelon form
b) Every pivot equals 1.
c) Each pivot is the only nonzero entry in its column.
example 0.8. example 0.9. Find the solution to the system
2x1 − x2 − 3x3 = 4
x1 + 2x2 + 2x3 = 6
3x1 + x2 + x3 = 8 4 example 0.10. Find the solution to the system
x1 + 2x2 + 2x3 = 6
2x1 − x2 − 3x3 = 4
3x1 + x2 − x3 = 8 5 example 0.11. Find the solution to the system
x1 + 2x2 + 2x3 = 6
2x1 − x2 − 3x3 = 4
3x1 + x2 − x3 = 10 definition 0.12. Consistent Linear System.
A linear system having one
or more solutions is called a consistent system. If it has no solutions, then it
is called an inconsistent system.
definition 0.13. The Gauss reduction with back substitution method
refers to the method of solving the system Ax = b by reducing [A|b] to a
row echelon form and then using back substitution.
The Guass-Jordan method refers to the method of solving the system
Ax = b by reducing [A|b] to a reduced row echelon form. 6 ...

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