Introduction to Matrices and Linear Systems in Engineering Analysis

1. Fundamentals of Engineering Analysis EGR 1302 - Introduction to Matrices

2. y y x x z a1x1 + b1x2 + c1x3 + d1x4 = e1 a2x1 + b2x2 + c2x3 + d2x4 = e2 a3x1 + b3x2 + c3x3 + d3x4 = e3 a4x1 + b4x2 + c4x3 + d4x4 = e4 y – mx = b rearranged to be ax + by = d Linear Systems y=mx+b OR: ax +by+cz = d

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

4. A = Basics of Matrix Notation Denoted by Capital Letters A, B, C … A Matrix is referred to by Row first, then column. 1 2 3 4 5 6 7 8 9 5 4 3 9 8 7 6 5 4 3 5 7 2 4 6 Row - Column m = # rows n = # columns A is an “m by n” or “m x n” matrix This matrix A is a 4x6 4x6 is the “Order” of the matrix

5. 1 2 3 4 5 5 3 4 1 7 8 3 5 6 0 2 1 7 9 8 7 6 5 4 0 3 5 7 4 2 B = A = 1 2 3 4 5 6 7 8 9 5 4 3 9 8 7 6 5 4 3 5 7 2 4 6 Elements of a Matrix Each element is denoted by lower case aij i row, j column a11 = 1 b34 = 6

6. 1 2 3 4 C= a column matrix [x,y,z] Order of Matrices B= [1 2 3 4] a row matrix A= [3]1x1 a scalar A column matrix is a “m X 1” A row matrix is a “1 X n” B= [1 2 3 4] is a “1 X 4” row matrix Row or column matrices are also referred to a “Vectors” A vector has magnitude and direction: The coordinates of a vector are represented with a matrix

7. 3 1 4 2 0 5 6 4 2 A = a12 = 1 a21 = 2 The Square Matrix All matrices are “rectangular”, but … When “m = n”, the matrix is “Square” or “n X n” “A” is a “3 X 3” square matrixx

8. a b c d 2 x 4 z A = B = 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 aij = bij When A = B a = 2 b = x c = 4 d = z

9. -3 2 1 4 k = 2, and A = -6 4 2 8 2A = 5 10 15 20 Factoring: if C = 1 2 3 4 Is not “C”! Also C = 5 * Basic Rules of Matrices (cont.) 2. Multiply a matrix by a constant Given k*A, where

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

11. 2 2 5 5 2 -1 3 6 0 3 2 -1 A = B = C = Basic Rules of Matrices (cont.) • Adding and Subtracting Matrices - Must be of the same Order = C, only if A + B A is a “m x n” and B is a “m x n” then C is a “m x n” cij aij + bij = A+B = Subtraction: (A – B) is the same as A+ (-1)*B

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

13. a11 a12 a21 a22 A = 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” Must be same Order A + B A = B if all respective elements are equal, and same Order aij Element denoted by lower case

14. Questions?