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Curl of a Vector Field
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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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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 Matrix Multiplication
Properties of Matrix Multiplication
Let
A, B
and
C
be matrices of dimensions such that the following are defined. Then
Associative:
A(BC) = (AB)C
Distributive:
A(B + C) = AB + AC
Distributive:
(A + B)C = AC + BC
c(AB) = (cA)B = A(cB)
, where c is a constant, please notice that A∙B ≠ B∙A
Multiplicative Identity:
For every square matrix
A
, there exists an identity matrix of the same order such that
IA = AI =A
.
Example 1:
Verify the associative property of matrix multiplication for the following matrices.
Solution:
Here we need to calculate both R.H.S (righthandside) and L.H.S (lefthandside) of
A (BC) = (AB) C
using (associative) property.
On the RHS we have:
and
On the LHS we have:
and
Hence the associative property is verified.
Example 2:
Verify the distributive property of matrix multiplication for the following matrices.
Solution:
Here we need to calculate both R.H.S and L.H.S of
A(B+C) = AB+AC
(distributive) property.
Hence the property is verified.
Example 3:
Find (
AB
)
C
and
A
(
BC
).
(AB)C
is determined as:
A(BC)
is determined as:
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