Matrix Division

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. 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

10. 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 AA-1 = I

11. 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 AA-1 = A-1 A = I However, for a square matrix A there is not always an inverse A-1

12. CW Inverse Matrix In matrices, the inverse of A is A-1 A-1 is defined so that AA-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. CW Inverse Matrix In matrices, the inverse of A is A-1 A-1 is defined so that AA-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 AA-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-1B

15. CW Inverse Matrix A-1 is defined so that AA-1 = A-1 A = I If A-1 does exist the the solution to AX=B is AX = B Pre-multiply by A-1 A-1AX = A-1B

16. CW Inverse Matrix A-1 is defined so that AA-1 = A-1 A = I If A-1 does exist the the solution to AX=B is AX = B Pre-multiply by A-1 A-1AX = A-1B ButA-1A = I so IX = A-1B X = A-1B

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

22. 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

23. CW General Inverse Matrix

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