Matrix Division

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CW
Matrix Division
We have seen that for 2x2 (two by two) matrices
A and B then AB ? BA To divide matrices we need
to define what we mean by division!
2
CW
Matrix Division
We have seen that for 2x2 (two by two) matrices
A and B then AB ? BA To divide matrices we need
to define what we mean by division! With numbers
or algebra we use b/a to solve axb. The
equivalent in 2x2 matrices is to solve AXB where
A, B and X are 2x2 matrices.
3
CW
Identity Matrix
With numbers or algebra we use b/a to solve axb.
The equivalent in 2x2 matrices is to solve AXB
where A, B and X are 2x2 matrices. We first need
to define the identity matrix - the matrix I for
which IX XI X for all X (For multiplying
number the identity is
4
CW
Identity Matrix
With numbers or algebra we use b/a to solve axb.
The equivalent in 2x2 matrices is to solve AXB
where A, B and X are 2x2 matrices. We first need
to define the identity matrix - the matrix I for
which IX XI X for all X (For multiplying
number the identity is 1). The identity 2x2
matrix is
5
CW
Identity Matrix
With numbers or algebra we use b/a to solve axb.
The equivalent in 2x2 matrices is to solve AXB
where A, B and X are 2x2 matrices. We first need
to define the identity matrix - the matrix I for
which IX XI X for all X (For multiplying
number the identity is 1). The identity 2x2
matrix is
6
CW
Identity Matrix
With numbers or algebra we use b/a to solve axb.
The equivalent in 2x2 matrices is to solve AXB
where A, B and X are 2x2 matrices. We first need
to define the identity matrix - the matrix I for
which IX XI X for all X (For multiplying
number the identity is 1). The identity 2x2
matrix is The identity 3x3 matrix is
7
CW
Identity Matrix
We first need to define the identity matrix - the
matrix I for which IX XI X for all X (For
multiplying number the identity is 1). The
identity 2x2 matrix is The identity 3x3 matrix
is In general if X is an mxn matrix then ImX
XIn X
8
CW
Identity Matrix
We first need to define the identity matrix - the
matrix I for which IX XI X for all X (For
multiplying number the identity is 1). The 2x2
identity matrix (I2) is The 3x3 identity matrix
(I3)is In general if X is an mxn matrix then
ImX XIn X
9
CW
Inverse Matrix
In numbers, the inverse of 3 is 1/3 3-1 In
algebra, the inverse of a is 1/a a-1 In
matrices, the inverse of A is A-1
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CW
Inverse Matrix
In numbers, the inverse of 3 is 1/3 3-1 In
algebra, the inverse of a is 1/a a-1 In
matrices, the inverse of A is A-1 3-1 is defined
so that 3x 3-1 1 a-1 is defined so that a x a-1
1 A-1 is defined so that A A-1 I
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CW
Inverse Matrix
In numbers, the inverse of 3 is 1/3 3-1 In
algebra, the inverse of a is 1/a a-1 In
matrices, the inverse of A is A-1 3-1 is defined
so that 3 x 3-1 3-1 x 3 1 a-1 is defined so
that a x a-1 a-1 x a 1 A-1 is defined so that
A A-1 A-1 A I However, for a square matrix A
there is not always an inverse A-1
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CW
Inverse Matrix
In matrices, the inverse of A is A-1 A-1 is
defined so that A A-1 A-1 A I However, for a
square matrix A there is not always an inverse
A-1 If A-1 does not exist then the matrix is
said to be singular If A-1 does exist then the
matrix is said to be non-singular
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CW
Inverse Matrix
In matrices, the inverse of A is A-1 A-1 is
defined so that A A-1 A-1 A I If A-1 does
not exist then the matrix is said to be
singular If A-1 does exist then the matrix is
said to be non-singular A square matrix A has an
inverse if, and only if, A is non-singular.
14
CW
Inverse Matrix
In matrices, the inverse of A is A-1 A-1 is
defined so that A A-1 A-1 A I A square
matrix A has an inverse if, and only if, A is
non-singular. If A-1 does exist the the
solution to AXB is X A-1 B
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CW
Inverse Matrix
A-1 is defined so that A A-1 A-1 A I If A-1
does exist the the solution to AXB is AX
B Pre-multiply by A-1 A-1AX A-1B
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Inverse Matrix
A-1 is defined so that A A-1 A-1 A I If A-1
does exist the the solution to AXB is AX
B Pre-multiply by A-1 A-1AX A-1B But
A-1A I so IX A-1B X A-1B
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Inverse Matrix
AX B Pre-multiply by A-1 A-1AX
A-1B But A-1A I so IX A-1B X
A-1B If the inverse of A is A-1 then the inverse
of A-1 is A. This is because if AC I then CA
I, and also any matrix inverse is unique.
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CW
Inverse Matrix
If the inverse of A is A-1 then the inverse of
A-1 is A. This is because if AC I then CA I,
and also any matrix inverse is unique. What is
the inverse of
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CW
Inverse Matrix
If the inverse of A is A-1 then the inverse of
A-1 is A. This is because if AC I then CA I,
and also any matrix inverse is unique. What is
the inverse of
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CW
Inverse Matrix
If the inverse of A is A-1 then the inverse of
A-1 is A. This is because if AC I then CA I,
and also any matrix inverse is unique. What is
the inverse of Then solve for u, v, w, x
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CW
General Inverse Matrix
If the inverse of A is A-1 then the inverse of
A-1 is A. This is because if AC I then CA I,
and also any matrix inverse is unique. What is
the inverse of
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CW
General Inverse Matrix
If the inverse of A is A-1 then the inverse of
A-1 is A. This is because if AC I then CA I,
and also any matrix inverse is unique. What is
the inverse of
Then solve for u, v, w, x
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CW
General Inverse Matrix
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General Inverse Matrix
What is the inverse of
Then solve for u, v, w, x