1 / 23

ENGR-1100 Introduction to Engineering Analysis

ENGR-1100 Introduction to Engineering Analysis. Lecture 11. Matrix and Matrix operation Rules of matrix arithmetic. Lecture Outline. 2 2 -2 0 -1 -1. 2 2 p -2 0 sin(2) -1 -1 e. 2 2 p -3. Definition: A matrix is a rectangular array of numbers.

signe-harding
Download Presentation

ENGR-1100 Introduction to Engineering Analysis

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. ENGR-1100 Introduction to Engineering Analysis Lecture 11

  2. Matrix and Matrix operation Rules of matrix arithmetic Lecture Outline

  3. 2 2 -2 0 -1 -1 2 2 p -2 0 sin(2) -1 -1 e 2 2 p -3 Definition: A matrix is a rectangular array of numbers The size of the matrix nxm. Where n the number of rows and m the number of columns.

  4. 2 2 -2 0 -1 -1 A= C= a11 a12….… a1n a21 a22….… a2n : : : an1 an2….… ann 2 2 p -3 A= • Capital letters denote matrices • Lower case letters denote numerical quantities In a square matrix m=n a11, a22….… ann are on the main diagonal

  5. Matrix are said to be equal if they have the same size, and the corresponding entries in the two matrices are equal. 2 2 -2 0 -1 -1 1 2 -2 0 3 -1 2 2 1 -2 0 2 -1 -1 2 A= B= C= A=B; B=C; A=C Definition 1: Equal Matrices

  6. 2 2 -2 0 -1 -1 1 2 -2 0 3 -1 2 2 1 -2 0 2 -1 -1 2 A= B= C= 3 4 -4 0 2 -2 Definition 2: Sum of Matrices • If A and B are any two matrices of the same size, then the sum A+B is the matrix obtained by adding together the corresponding entries in the two matrices. Matrices of different sizes can’t be added. A+C and B+C are undefined A+B=

  7. 2 2 -2 0 -1 -1 4 4 -4 0 -2 -2 A= -2 -2 2 0 1 1 (-1)A=-A= Definition 3: Matrix Scalar Product • If A is any matrix and c any scalar, then the product c.A is the matrix obtained by multiplying each entry of A by c. 2A=

  8. 1 4 3 • 0 -1 3 1 • 2 7 5 2 1 2 4 2 6 0 A= B= Definition 4: Product of two matrices A . B =?

  9. The size of a product matrix A B A.B = m x rr x nm x n inside outside

  10. 4 0 2 1 -1 7 4 3 5 3 1 2 1 2 4 = 2 6 0 26 c14 4 0 2 1 -1 7 4 3 5 3 1 2 13 1 2 4 = 2 6 0 26 (1x3)+(2x1)+(4x2)=13 2 x 4 3 x 4 2 x 3 c23 = a21xb13+a22xb23+a23xb33 (2x4)+(6x3)+(0x5)=26

  11. 4 0 2 1 -1 7 4 3 5 3 1 2 13 1 2 4 = 2 6 0 26 12 27 30 13 A.B = 8 -4 26 12 Class Assignment: Complete the product computation

  12. 3 0 -1 2 1 1 4 -1 0 2 3 x 2 2 x 2 B= Matrix size 3 x 2 A= 12 -3 -4 5 4 1 inside A.B= outside Class Assignment Problem:Set 1.4-4a: determineA.B

  13. 1 5 2 -1 0 1 3 2 4 6 1 3 -1 1 2 4 1 3 D= E= 3 x 3 3 x 3 Matrix size 3 x 3 inside outside 9 8 19 -2 0 0 32 9 25 D.E= Class Assignment Problem:Set 1.4-4d: Determine D.E

  14. 1 5 2 -1 0 1 3 2 4 6 1 3 -1 1 2 4 1 3 D= E= 3 x 3 3 x 3 Matrix size 3 x 3 inside outside 14 36 25 4 -1 7 12 26 21 D.E= Class Assignment Problem:Set 1.4-4e: DetermineE.D

  15. 6 5 • -2 1 3 • 7 3 7 D+E= Class Assignment Problem:Set 1.4-4b: DetermineD+E 1 5 2 -1 0 1 3 2 4 6 1 3 -1 1 2 4 1 3 D = E =

  16. -1 0 2 3 1 2 3 0 A= B= -1 -2 11 4 3 6 -3 0 A.B= B.A= Therefore: A.B=B.A Rules of Matrix Arithmetic Multiply A.B: Multiply B.A

  17. (1) A+B=B+A (2) A+(B+C)=(A+B)+C (3) A(BC)=(AB)C (4) A(B+C)=AB+AC (5) (B+C)A=BA+CA (6) A(B-C)=AB-AC (7) (B-C)A=BA-CA (8) a(B+C)=aB+aC (9) a(B-C)=aB-aC (10) (a+b)C=aC+bC (11) (a-b)C=aC-bC (12) (ab)C=a(bC) (13) a(BC)=(aB)C=B(aC) The following rules of matrix arithmetic are valid(assuming that the sizes of the matrices are such that the indicated operations can be performed)

  18. Transpose of a Matrix, At a11 a12 a13 a14 a21 a22 a23 a24 A= a31 a32 a33 a34 a11 a12 a13 a14 a21 a22 a23 a24 a31 a32 a33 a34 At=

  19. 1 0 0 1 I2= 1 0 0 0 I3= 0 1 0 0 0 0 1 0 0 0 0 1 I4= 1 0 0 0 1 0 0 0 1 Identity Matrix:A square matrix with 1’s on the main diagonal and 0’s off the main diagonal If A is an mxn matrix, then: AIn=A and ImA=A

  20. 3 5 1 2 3 5 1 2 2 -5 -1 3 2 -5 -1 3 1 0 0 1 1 0 0 1 2 -5 -1 3 A= A.B= = = I B.A= = Inverse of a MatrixIf Ais a square matrix, and a matrixBissuch thatA.B=B.A=I, thenAis said to be invertible andBis called the inverse ofA 3 5 1 2 is the inverse of B=A-1= = I

  21. x(-1) x(-1) a b c d d -b -c a A= 1 ad-bc A-1= = d ad-bc -b ad-bc -c ad-bc a ad-bc How to find the inverse matrix? For a 2x2 matrix If ad-bc=0 then

  22. Using the formula: 3 -2 -1 1 1 2 1 3 d -b -c a d -b -c a A= Since: ad-bc=3-2=1 1 ad-bc A-1= = A-1= = -c ad-bc a ad-bc -b ad-bc d ad-bc Find the inverse of: a=1; b=2; c=1; d=3

  23. 3 4 5 6 A-1= Class Assignment Problem:Set 1.5-6: Let A be an invertable matrix whose inverse is: Find the matrix A

More Related