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Curl of a Vector Field
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Divergence of a Vector Field
·
Gradient of a Scalar Field
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Properties of Transposes
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The Transpose of a Matrix
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Cartesian Coordinate
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·
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·
Transform from Cartesian to Cylindrical Coordinate
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·
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·
Divergence Theorem/Gauss' Theorem
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Stokes' Theorem
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Definition of a Matrix
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>
Math Formulas
>
Linear Algebra
> Properties of Determinants
Properties of Determinants
Properties of Determinants:
· Let
A
be an
n × n
matrix and
c
be a scalar then:
· Suppose that
A
,
B
, and
C
are all
n × n
matrices and that they differ by only a row, say the
k
^{th}
row. Let’s further suppose that the
k
^{th}
row of
C
can be found by adding the corresponding entries from the
k
^{th}
rows of
A
and
B
. Then in this case we will have:
The same result will hold if we replace the word row with column above.
· If
A
and
B
are matrices of the same size then:
· Suppose that
A
is an invertible matrix then:
· A square matrix
A
is invertible if and only if
det(
A
) ≠ 0.
A
matrix that is invertible is often called
nonsingular
and a matrix that is not invertible is often called
singular
.
· If
A
is a square matrix then:
· If
A
is a square matrix with a row or column of all zeroes then:
det(
A
) = 0 and so
A
will be singular.
· Suppose that
A
is an
n × n
triangular matrix then:
· If two rows (or columns) are interchanged, the sign of the determinant is changed.
Example 1
: For the given matrix below compute both det(
A
) and det(2
A
).
Also verify the property det(c
A
) = c
^{n}
det(
A
).
Solution:
First of all, we’ll find the scalar multiples of the given matrix.
The determinants:
det(
A
) = 45
det(2
A
) = 360 = (8)(45) = 2
^{3}
det(
A
)
Hence the property is verified.
Example 2:
Let
A
be an
n × n
matrix.
(a) det(
A
) = det(
A
^{T}
)
(b) If two rows (or columns) of
A
are equal, then det(
A
) = 0.
(c) If a row (or column) of
A
consists entirely of 0, then det(
A
) = 0.
Verify the above properties of determinants for the following matrices:
Solution:
Property (a) holds
Property (b) holds.
Property (c) holds.
Example 3
:
Consider the following three matrices.
Verify that det(
C
) = det(
A
) + det(
B
)
Solution: First, notice that we can write
C
as:
All three matrices differ only in the second row and the second row of
C
can be found by adding the corresponding entries from the second row of
A
and
B
.
The determinants of these matrices are:
det(
A
) = 15
det(
B
) = 115
det(
C
) = 100 = 15 + (115)
Hence the property is verified.
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