1. Chapter 8 Matrices and Determinants By Richard Warner, Nate Huyser, Anastasia Sanderson, Bailey Grote

2. Chapter 8.1: General Matrices Rectangular array of numbers called entries Dimensions of a matrix are number of rows by the number of columns

3. 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

4. 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 A 13 is the x-coordinate of the solution A 23 is the y-coordinate of the solution

5. 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 A 11 = 1 3.Work down, around, and up to achieve RREF 4.Write last column as ordered pair for final answer

6. Chapter 8.1: Solving with Calculator (RREF) Only used for Matrices larger than 2x2 1.(2 nd ) [Matrix] → EDIT 2.Matrix[A] 3x4 3.Enter entries by rows 4.(2 nd ) [Quit] 5.(2 nd ) [Matrix] → MATH 6.Select [RREF] 7.(2 nd ) [Matrix] select Martix[A]

7. 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

8. 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 A mxn * B nxr =AB mxr Ex: p.598 #29

9. Identity Matrices I 2x2 I 3x3 8.3 Inverse Matrices

10. A A -1 =A -1 A =I If A= where ad-bc cannot equal 0, Then A -1 =1/(ad-bc) * Inverse of2x2: Cont. 8.3 Inverse Multiplication

11. 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=A 1 B x= Cont. 8.3 Inverse Multiplication

12. 8.4 Determinants a real number derived from a square matrix If A = then Det[A]= AD-CB For 2x2 matrices only For 3x3 matrices or larger 1.(2 nd ) Matrix → [Edit] A 2.Enter dimensions 3.(2 nd ) Quit 4.(2 nd ) Matrix → [Math] enter 5.(2 nd ) Matrix → enter

13. 8.5 Determinant Applications Cramer’s Rule solves systems using determinates. Example:

14. 8.5 Determinant Applications Finding the area of a triangle where the points are (a,b), (c,d), (e,f) Points are collinear if A=0