© 2005 Baylor University Slide 1 Fundamentals of Engineering Analysis EGR Introduction to Matrices Approximate Running Time - 12 minutes Distance Learning / Online Instructional Presentation Presented by Department of Mechanical Engineering Baylor University Procedures: 1.Select “Slide Show” with the menu: Slide Show|View Show (F5 key), and hit “Enter” 2.You will hear “CHIMES” at the completion of the audio portion of each slide; hit the “Enter” key, or the “Page Down” key, or “Left Click” 3.You may exit the slide show at any time with the “Esc” key; and you may select and replay any slide, by navigating with the “Page Up/Down” keys, and then hitting “Shift+F5”.

© 2005 Baylor University Slide 2 y=mx+b x y x y z y – mx = b rearranged to be ax + by = d OR: ax +by+cz = d Linear Systems a 1 x 1 + b 1 x 2 + c 1 x 3 + d 1 x 4 = e 1 a 2 x 1 + b 2 x 2 + c 2 x 3 + d 2 x 4 = e 2 a 3 x 1 + b 3 x 2 + c 3 x 3 + d 3 x 4 = e 3 a 4 x 1 + b 4 x 2 + c 4 x 3 + d 4 x 4 = e 4

© 2005 Baylor University Slide 3 How Do We Manage Large Amounts of Data? Matrix Algebra We arrange data in a: Matrix = Table = Array The key is learning the Definitions Symbology Notation

© 2005 Baylor University Slide 4 Basics of Matrix Notation Denoted by Capital Letters A, B, C … A = A Matrix is referred to by Row first, then column. Row - Column m = # rowsn = # columnsA is an “m by n” or “m x n” matrix This matrix A is a 4x64x6 is the “Order” of the matrix

© 2005 Baylor University Slide 5 B = A = Elements of a Matrix Each element is denoted by lower case a ij i row, j column a 11 = 1 b 34 = 6

© 2005 Baylor University Slide 6 Order of Matrices A= [3] 1x1 a scalar B= [ ] a row matrix C= a column matrix A row matrix is a “1 X n” A column matrix is a “m X 1” B= [ ] is a “1 X 4” row matrix Row or column matrices are also referred to a “Vectors” A vector has magnitude and direction: [x,y,z] The coordinates of a vector are represented with a matrix

© 2005 Baylor University Slide 7 The Square Matrix All matrices are “rectangular”, but … When “m = n”, the matrix is “Square” or “n X n” A = “A” is a “3 X 3” square matrixx a 21 = 2 a 12 = 1

© 2005 Baylor University Slide 8 Basic Rules of Matrices 1. Equality – two matrices are equal if - They are both the same “order” - All respective elements are equal In other words a ij = b ij A = a b c d B = 2 x 4 z a = 2 b = x c = 4 d = z When A = B

© 2005 Baylor University Slide 9 Basic Rules of Matrices (cont.) 2. Multiply a matrix by a constant Given k*A, where k = 2, and A = A = Factoring: if C = Also C = 5 * Is not “C”!

© 2005 Baylor University Slide 10 Basic Rules of Matrices (cont.) 3. The Null Matrix - All elements are Zero A = 0 A is a Null Matrix

© 2005 Baylor University Slide 11 Basic Rules of Matrices (cont.) 4.Adding and Subtracting Matrices - Must be of the same Order A + B = C, only if A is a “m x n” and B is a “m x n”then C is a “m x n” a ij + b ij = c ij A = B = A+B = C = 2 5 Subtraction: (A – B) is the same as A+ (-1)*B

© 2005 Baylor University Slide 12 Basic Rules of Matrices (cont.) 5. Associative Law (A + B) + C = A + (B + C) k*(A + B) = k*A + k*B Now for a review of this lesson -

© 2005 Baylor University Slide 13 Review of Matrix Rules - Table or Array - Capital Letters – “A” - Rectangular or Square - Order: m x n, or m=n is square ( n x n) - m = #rows, n = #columns – always “row-column” A + B Must be same Order A = B if all respective elements are equal, and same Order Element denoted by lower case a ij A = a 11 a 12 a 21 a 22

© 2005 Baylor University Slide 14 This concludes the Lecture