Section 2.1 Operations with Matrices

1. Section 2.1Operations with Matrices

2. Notation: If A is a matrix with m rows and n columns, then we say A is an mxn matrix and we can write: A = [ aij ]

3. Terminology: Column matrix (column vector) – A matrix with just one column. Row matrix (row vector) – A matrix with just one row.

4. Equality of Matrices Two matrices A and B are said to be equal if aij = bij for all possible i and j.

5. Addition/Subtraction of Matrices, Scalar Multiplication Ex. Let (a) 2A = (b)A + B = (c)A –2B =

6. Addition/Subtraction of Matrices, Scalar Multiplication Ex.

7. Matrix Multiplication Let You might think that AB = Actually, AB =

8. Matrix Multiplication Dot Products = a1b1 + a2b2 + a3b3 + . . . + anbn

9. Matrix Multiplication Dot Products = a1b1 + a2b2 + a3b3 + . . . + anbn Note: In a dot product the row matrix must come first and the column matrix must come second.

10. Matrix Multiplication Examples: = = (3)(2) + (1)(1) + (–4)(5) + (2)(7) = 1 (2)(1) + (1)(2) + (5)(3) = 19

11. Matrix Multiplication Some dot products don't make sense. = (3)(2) + (1)(1) + (–4)(5) + (2)( )

12. Matrix Multiplication To compute AB, we must multiply each row in A with each column in B. If A is an mxn matrix and B is an nxp matrix, then C = AB is an mxp matrix and cij = ai1b1j+ ai2b2j+ ai3b3j+ . . . + ainbnj

13. Matrix Multiplication AB 22 2 3 50

14. Matrix Multiplication AB –6 6 9 2 4 3 4 –7 3

15. Matrix Multiplication Ex. Determine if the following matrix multiplications can be done. If they can, give the dimensions of the resulting matrix. But do not actually perform the multiplication. (a)

16. Matrix Multiplication Ex. Determine if the following matrix multiplications can be done. If they can, give the dimensions of the resulting matrix. But do not actually perform the multiplication. (b)

17. Matrix Multiplication Ex. Determine if the following matrix multiplications can be done. If they can, give the dimensions of the resulting matrix. But do not actually perform the multiplication. (c)

18. Matrix Multiplication Ex. Determine if the following matrix multiplications can be done. If they can, give the dimensions of the resulting matrix. But do not actually perform the multiplication. (d)

19. Matrix Multiplication Ex. Determine if the following matrix multiplications can be done. If they can, give the dimensions of the resulting matrix. But do not actually perform the multiplication. (e)

20. Matrix Multiplication Ex. Use a calculator to perform the following matrix multiplication.

21. Matrix Multiplication Ex. Let and . Compute AB and then compute BA.

22. Matrix Multiplication A system of many linear equations can be handled by one matrix equation: AX = B. 3x – y – z = 5 4x + 2y + 6z = –2 → 5x + 3y – 8z = 7

23. Matrix Multiplication Solve AX = B for X.

24. Ex. Solve I + X = A for X, where I = and A =

25. A system of linear equations can be written as a linear combination of column vectors. Ex. Write the following system as a linear combination of column vectors: 3x – 2y + z = 3 –x + 3y + 2z = 4 x– y = 3

26. Section 2.2Properties of Matrix Operations

27. Def. The transpose of an mxn matrix A, denoted AT, is an nxm matrix defined by the following: If A = [aij] then AT = [ aji ].

28. Ex. Compute the transpose of the following matrices.

29. Def. A matrix A is symmetric if A = AT.

30. Recall that for matrices A and B, it is not always true that AB = BA. Another property which is not always true for matrices (but we’d like to be true): If AC = BC then A = B.

31. Ex. Let , , and . Compute AC and BC. Verify that AC = BC, but A ≠ B.

32. Properties Let A, B, and C be matrices and let s and t be scalars. 1. A + B = B + A 2. A + (B + C) = (A + B) + C 3. s(tA) = (st)A 4. s(A + B) = sA + sB 5. (s + t)A = sA + tA 6. A(BC) = (AB)C 7. A(B + C) = AB + AC 8. (A + B)C = AC + BC

33. The zero matrix - O

34. The identity matrix - I

35. Properties Let A, B, and C be matrices and let s and t be scalars. 9. A + O = A 10. A + (-A) = O 11. If sA = O then either s = 0 or A = O 12. AI = A 13. IA = A

36. Ex. Verify number 12 and 13 above for

37. Another property which is not always true for matrices: If AB = O (the zero matrix) then A = O or B = O.

38. Ex. Let and Verify that AB = O, but A ≠ O and B ≠ O

39. Powers of a (square) matrix. Ak = A·A·A·····A (k factors of A multiplied together).

40. Ex. Let A2 = A3 = A39 =

41. Ex. Let A0 = (guess) A-1 = (guess)

42. Section 2.3The Inverse of a Matrix

43. For scalars, a-1 = 1/a . The actual definition of a-1 is: If a ≠ 0 then there is a unique number denoted a-1 so that a · a-1 = 1 and a-1 · a = 1.

44. For scalars, a-1 = 1/a . The actual definition of a-1 is: If a ≠ 0 then there is a unique number denoted a-1 so that a · a-1 = 1 and a-1 · a = 1. If A is a square matrix with ____ , then there is a unique matrix denoted A-1 so that A · A-1 = I and A-1 · A = I.

45. Terminology If A-1 exists, then we say that A is invertible (non-singular). If no such A-1 exists then we say that A is singular (non-invertible).

46. Ex. Let and . Show that B = A-1.

47. Solve each of the following systems of equations, then do so all at once. 2x –4y = –6 2x –4y = 10 3x + y = 5 3x + y = 8 1/2R1→R1 1/2R1→R1 R2 –3R1→R2 R2 –3R1→R2 1/7R2→R2 1/7R2→R2 R1 +2R2→R1 R1 +2R2→R1 (1, 2) (3, 1)

48. Solving both at once: Solving each separately: 1/2R1→R1 1/2R1→R1 R2 –3R1→R2 R2 –3R1→R2 1/7R2→R2 1/7R2→R2 R1 +2R2→R1 R1 +2R2→R1 (3, 1) (1, 2) (1, 2) and (3, 1)

49. Now back to computing A-1.

50. Let's compute A-1 if we have Let Our job is to determine the values of a, b, c, d. Now, Multiply the two matrices on the left to get: Since the two matrices are equal, the corresponding components of these two matrices must be equal. This gives us four equations in four unknowns: 2a + c = 1 3a + 2c = 0 2b + d = 0 3b + 2d = 1 We can separate this into two sets of two equations with two unknowns: 2a + c = 1 2b + d = 0 3a + 2c = 0 3b + 2d = 1