matrix division
DESCRIPTION
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!. CW. Matrix Division. We have seen that for 2x2 (“two by two”) matrices A and B then AB BA - PowerPoint PPT PresentationTRANSCRIPT
Matrix Division
CW
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!
Matrix Division
CW
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 ax=b. The equivalent in 2x2 matrices is to solve AX=B where A, B and X are 2x2 matrices.
Identity MatrixCW
With numbers or algebra we use b/a to solve ax=b. The equivalent in 2x2 matrices is to solve AX=B 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
Identity MatrixCW
With numbers or algebra we use b/a to solve ax=b. The equivalent in 2x2 matrices is to solve AX=B 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
Identity MatrixCW
With numbers or algebra we use b/a to solve ax=b. The equivalent in 2x2 matrices is to solve AX=B 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
1001
Identity MatrixCW
With numbers or algebra we use b/a to solve ax=b. The equivalent in 2x2 matrices is to solve AX=B 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
1001
100010001
Identity MatrixCW
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
1001
100010001
Identity MatrixCW
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
1001
100010001
Inverse MatrixCW
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
Inverse MatrixCW
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 = 1a-1 is defined so that a x a-1 = 1A-1 is defined so that A A-1 = I
Inverse MatrixCW
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 = 1a-1 is defined so that a x a-1 = a-1 x a = 1A-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
Inverse MatrixCW
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
Inverse MatrixCW
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.
Inverse MatrixCW
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 AX=B is
X = A-1 B
Inverse MatrixCW
A-1 is defined so that A A-1 = A-1 A = I
If A-1 does exist the the solution to AX=B is
AX = BPre-multiply by A-1 A-1AX = A-1B
Inverse MatrixCW
A-1 is defined so that A A-1 = A-1 A = I
If A-1 does exist the the solution to AX=B is
AX = BPre-multiply by A-1 A-1AX = A-1B
But A-1A = I so IX = A-1B X = A-1B
Inverse MatrixCW
AX = BPre-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.
Inverse MatrixCW
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
3002
A
Inverse MatrixCW
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
3002
A
yx0
0let 1A
Inverse MatrixCW
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
3012
B
xwvu1let B
2013
611B
General Inverse MatrixCW
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
dcba
C
General Inverse MatrixCW
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
dcba
C
xwvu1let C
Then solve for u, v, w, x
General Inverse MatrixCW
dcba
C
bcadDacbd
Dxwvu
where
1let 1C
1001
dxcvbxavdwcubwau
ac
cwbcadSubtract
dawcaucbcwacu
)(:
0
General Inverse MatrixCW
What is the inverse of
363121
31
6321
31
6321
yxyx
yxyx
yx
xwvu1let C
Then solve for u, v, w, x