Chapter 7 Notes Honors Pre-Calculus

7.1/7.2 Solving Systems Methods to solve: EXAMPLES: Possible intersections: 1 point, 2 points, none Elimination, Substitution, Calculator MONEY: Invest $12,000 into two accounts, 9% and 11% interest. Yearly interest is $1180. How much in each? Calculator Problems

7.2 Examples Special Cases: Consistent: Intersect or Coincide Inconsistent: Parallel, No intersection X, total - X

7.3 Multivariable Systems 3 Variables – Ordered Triple Answers GOAL: Get to two equations, two unknowns. Examples: Infinite Solutions: (a’s) True statements and non-square systems 3-D Coordinate System – Graph is a Plane

7.3 Partial Fractions Partial Fraction Decomposition: Decomposition into Partial Fractions: 1)Divide if Improper 2)Factor Denominator 3)Linear Factors 4)Repeated Factors Examples:

7.4 Matrices and Systems Matrices m X n matrix rows X columns Elementary Row Operations 1.Interchange 2 rows 2.Multiply a Row by a nonzero constant 3.Add a Multiple of one row to another. Writing an Augmented Matrix: System: 2 Matrices are possible from a system.

7.4 Cont’d. Coefficient Matrix Augmented Matrix Row – Echelon Form Solve the following system of equations with matrices.

7.5 Operations w/ Matrices Representations of Matrices p. 504 Matrices are equal if same order and all corresponding entries are equal. Addition of Matrices – Must be same order. Scalar Multiplication – each entry is mult. By the scalar. Additive Identity MatrixMultiplicative Identity To Multiply two matrices, A and B, the number of columns in A must equal the number of rows in B. Order of answer matrix is the rows in A by the columns in B.

7.5 cont’d. Multiply AB and BA, if possible………. A=B= AB=BA= Not Possible Identity Matrix (I) must be n X n AI = A or IA = A

7.6 Inverse of a Square Matrix Let A be an nXn matrix and let I be an nXn Identity matrix. Then the two matrices are Inverses. 2 ways to find an inverse for a 2 X 2 matrix. Option 1: Augment the matrix with the identity and turn your matrix into the identity. Option 2: Use the following formula: to find the inverse of 2 X 2…. The inverse of

7.7 Determinant of a Square Matrix Every square matrix can be associated with a real number called its determinant. DET A =If A= Examples: For a 3 X 3, augment the matrix with its first two columns and multiply your 3 down diagonals (add) and multiply your 3 up diagonals (add) THEN Down – Up.