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Curl of a Vector Field
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Cartesian Coordinate
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Stokes' Theorem
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Definition of a Matrix
Current Location
>
Math Formulas
>
Linear Algebra
> Elementary Matrices
Elementary Matrices
An
n
×n
matrix is called an
elementary matrix
if it can be obtained from the
n
×n
identity matrix
I
_{n}
by performing a
single
elementary row operation.
Properties of Elementary Matrices:
a. If the elementary matrix
E
results from performing a certain row operation on
I
_{m}
^{ }
and if
A
is an
m
×n
matrix, then the product
EA
is the matrix that results when this same row operation is performed on
A
.
b. Every elementary matrix is invertible, and the inverse is also an elementary matrix.
Example 1:
Give four elementary matrices and the operations that produce them.
Solution:
Listed below are four elementary matrices attached with the operations that produced them.
a.
Multiply the second row of
I
_{2}
by 3.
b.
Interchange the second and fourth rows of
I
_{4}
_{.}
c.
Add 3 times the third row of
I
_{3}
to the first row.
d.
Multiply the first row of
I
_{3}
by 1.
Example 2:
Verify first property
of elementary matrices for the following 3
×
4 matrix.
Solution:
Consider the matrix
and consider the elementary matrix
which results from adding 3 times the first row of
I
_{3}
to the third row. The product
EA
is
which is precisely the same matrix that results when we add 3 times the first row of
A
to the third row. Hence the property is verified.
Example 3:
What should we premultiply
by if we want to multiply row 3 by m?
Solution:
We start with the 4×4 identity matrix
, we then multiply row three by
m
to obtain
.This is the desired elementary matrix. We can check that if we multiply
A
by this matrix, the resulting matrix will be
A
in which row three has been multiplied by
m
.
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