1. 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 ax=b. The equivalent in 2x2 matrices is to solve AX=B where A, B and X are 2x2 matrices.

3. CW
Identity Matrix
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

4. CW
Identity Matrix
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

5. CW
Identity Matrix
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

6. CW
Identity Matrix
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

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. Inverse Matrix CW 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

10. Inverse Matrix CW 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

11. Inverse Matrix CW 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

12. Inverse Matrix CW 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

13. Inverse Matrix CW 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. Inverse Matrix CW 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

15. Inverse Matrix CW 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 = B
Pre-multiply by A A-1AX = A-1B

16. Inverse Matrix CW 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 = B
Pre-multiply by A A-1AX = A-1B
But A-1A = I so IX = A-1B
X = A-1B

17. Inverse Matrix CW 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.

18. 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

19. 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

20. 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

21. General Inverse MatrixCW
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

22. General Inverse MatrixCW
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

23. General Inverse MatrixCW
General Inverse Matrix

24. General Inverse MatrixCW
General Inverse Matrix
What is the inverse of
Then solve for u, v, w, x