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Transcript
Chapter 8 Matrices and Determinants By Richard Warner, Nate Huyser, Anastasia Sanderson, Bailey Grote Chapter 8.1: General Matrices • Rectangular array of numbers called entries • Dimensions of a matrix are number of rows by the number of columns A11 A21 A31 A12 A22 A32 A13 A23 A33 Chapter 8.1: Augmented Matrices • Augmented Matrix- derived from a system of equations • Elementary Row Operations • Interchange any two rows • Multiply any row by a nonzero constant • Add two rows together • 2x2 by hand, 3x3 with calculator Chapter 8.1: Reduced Row Echelon Form (RREF) • Any rows consisting of all zeros occur at the bottom of the matrix • All entries on the main diagonal are 1 • All entries not on the main diagonal or in the last column are 0 • A13 is the x-coordinate of the solution • A23 is the y-coordinate of the solution 1 0 0 1 x y Chapter 8.1: Gauss Jordan Elimination • Uses Augmented Matrices to solve systems of equations 1. Write system as an augmented matrix 2. Use the row operations to make A11 = 1 3. Work down, around, and up to achieve RREF 4. Write last column as ordered pair for final answer Chapter 8.1: Solving with Calculator (RREF) • 1. 2. 3. 4. 5. 6. 7. Only used for Matrices larger than 2x2 (2nd) [Matrix] → EDIT 1 0 0 Matrix[A] 3x4 Enter entries by rows 0 1 0 (2nd) [Quit] (2nd) [Matrix] → MATH 0 0 1 Select [RREF] (2nd) [Matrix] select Martix[A] x y z Chapter 8.2: Matrix Operations Equality of Matrices: 2 matrices are equal if they have the same dimensions and their corresponding entries are equal To add and subtract Matrices: They must have the same dimensions. •Add the corresponding entries Scalar Multiplication: •Multiplying a matrix by a scalar (constant) •Multiply each entry in the matrix by the scalar Chapter 8.2: Matrix Operations Matrix Multiplication: • To Multiply AB, A’s columns must equal B’s rows • Multiply the entries in A’s rows by the corresponding entries in B’s columns • Amxn* Bnxr =ABmxr Ex: p.598 #29 8.3 Inverse Matrices Identity Matrices I 2x2 1 0 0 1 I 3x3 1 0 0 0 1 0 0 0 1 Cont. 8.3 Inverse Multiplication Inverse of2x2: • A A-1 =A-1 A =I • If A= a b where ad-bc cannot equal 0, c d Then A-1 =1/(ad-bc) * d b c a Cont. 8.3 Inverse Multiplication Inverse of 3x3 1. Enter [matrix] in calculator 2. [matrix][A] [enter] [x-1 ] [enter] To solve a system of linear equations 1. Write the system of equations as a matrix problem 2. Find A-1 3. X=A1B x1 x2 x3 y1 y2 y3 z1 x z2 z 3 x y z = c1 c 2 c3 8.4 Determinants • a real number derived from a square matrix • If A = CA DB then Det[A]= AD-CB • For 2x2 matrices only • For 3x3 matrices or larger 1. 2. 3. 4. 5. (2nd) Matrix → [Edit] A Enter dimensions (2nd) Quit (2nd) Matrix → [Math] enter (2nd) Matrix → enter 8.5 Determinant Applications • Cramer’s Rule solves systems using determinates. Dy Dx x • Example: D y e b f d x a b c d D a e c f y a b c d 8.5 Determinant Applications • Finding the area of a triangle where the points are (a,b), (c,d), (e,f) a 1 c 2 e b 1 d 1A f 1 • Points are collinear if A=0 a c e b 1 d 1 0 f 1