4.1: Matrix Operations Objectives: Students will be able to: Add, subtract, and multiply a matrix by a scalar Solve Matrix Equations Use matrices to organize data

Matrix A rectangular arrangement of numbers in rows and columns Dimensions of a Matrix: # rows by # columns 2 X 3 (read 2 by 3)

Entries: the numbers in a matrix Square Matrix: a matrix with the same # of rows and columns

What are the dimensions of the matrices below? 1. 2. 3.

Are the following matrices equal? 1. and 2. and Two matrices are equal if their dimensions are the same and the entries in corresponding positions are equal. Are the following matrices equal? 1. and 2. and

To add and subtract matrices, add or subtract corresponding entries: Can only add and subtract if matrices have the same dimensions Perform the indicated operations: 1. 2. 3.

Scalar Multiplication: multiply each entry of the matrix by the scalar 1. 2.

Solve for x and y: 3x = -9, x = -3 3y-2 =7, y = 3

(A+B)+C = A +(B+C) A + B = B +A c(A +B) = cA + cB c(A- B) = cA- cB Properties of Matrix Operations: A, B and C are matrices, c is a scalar Associative Property (regroup) Commutative Property (change order Distributive Property of Addition Distributive Property of Subtractions (A+B)+C = A +(B+C) A + B = B +A c(A +B) = cA + cB c(A- B) = cA- cB

Using Matrices to Organize Data: Use matrices to organize the following data about insurance rates. This year for 1 car, comprehensive, collision and basic insurance cost $612.15, $518.29 and $486.91. For 2 cars, comprehensive, collision and basic insurance cost $1150.32, $984.16, and $892.51. Next year for 1 car, comprehensive, collision and basic insurance will cost $616.28, $520.39, and $490.05. For 2 cars, comprehensive, collision and basic insurance will cost $1155.84, $987.72, and $895.13.

Use the matrices to write a matrix that shows the changes from this year to next.

This year (A) Next year (B) 1 car 2 cars Comp. Coll. basic B – A will give the change of:

Multiplying Matrices -You can only multiply matrices when the number of columns in the first matrix is equal to the number of rows in the second. -Multiplication of matrices is not commutative!! -The dimensions of the product matrix will be the number of rows in the first matrix by the number of columns in the second matrix 3 x 2 matrix times a 2 x 2 matrix results in a 3 x 2 matrix

Multiply each row entry by each column entry to yield one entry in the product matrix. Must be the same Dimensions of product matrix

The determinant of a matrix is the difference in the cross products Determinants The determinant of a matrix is the difference in the cross products det A or lAl

Find the determinant of a 2 x 2 matrix

Using Diagonals Another method for evaluating a third order determinant is using diagonals. STEP 1: You begin by repeating the first two columns on the right side of the determinant.

Using Diagonals aei bfg cdh STEP 2: Draw a diagonal from each element in the top row diagonally downward. Find the product of the numbers on each diagonal. aei bfg cdh

Using Diagonals gec hfa idb STEP 3: Then draw a diagonal from each element in the bottom row diagonally upward. Find the product of the numbers on each . gec hfa idb

Using Diagonals To find the value of the determinant, add the products in the first set of diagonals, and then subtract the products from the second set of diagonals. The value is: aei + bfg + cdh – gec – hfa – idb

Ex. 2: Evaluate using diagonals. First, rewrite the first two columns along side the determinant.

Ex. 2: Evaluate using diagonals. Next, find the values using the diagonals. -5 24 4 60 Now add the bottom products and subtract the top products. 4 + 60 + 0 – 0 – (-5) – 24 = 45. The value of the determinant is 45.

Area of a triangle Determinants can be used to find the area of a triangle when you know the coordinates of the three vertices. The area of a triangle whose vertices have coordinates (a, b), (c, d), (e, f) can be found by using the formula: and then finding |A|, since the area cannot be negative.

Ex. 3: Find the area of the triangle whose vertices have coordinates (-4, -1), (3, 2), (4, 6). How to start: Assign values to a, b, c, d, e, and f and substitute them into the area formula and evaluate. a = -4, b = -1, c = 3, d = 2, e = 4, f = 6 8 -24 -3 18 -8 -4 Now add the bottom products and subtract the top products. -8 + (-4) + 18 – 8 – (-24) –(-3) = 25. The value of the determinant is 25. Applied to the area formula ½ (25) = 12.5. The area of the triangle is 12.5 square units.

Finding the inverse of a matrix (the determinant cannot be zero, if the determinant is zero the matrix has no inverse.

Try this one! Find the inverse of

1) Inverse Matrices and Systems of Equations For a We can write a System of Equations Matrix Equation

1) Inverse Matrices and Systems of Equations Example 1: Write the system as a matrix equation Matrix Equation Coefficient matrix Variable matrix Constant matrix

1) Inverse Matrices and Systems of Equations When rearranging, take the inverse of A

1) Inverse Matrices and Systems of Equations Example 3: Solve the system Step 3: Solve for the variable matrix The solution to the system is (4, 1).

1) Inverse Matrices and Systems of Equations Example 2: A X B (14.5, 8, -6.5)

Solving systems using Augmented Matrices You can solve some linear systems by using an augmented matrix. An augmented matrix contains the coefficients and the constants from a system of equations. Each row of the matrix represents an equation.

Ex. 1: Write an augmented matrix to represent the system shown. System of Equations -6x - 2y = 10 4x = -20 System of Equations Use the rref key under the matrix math menu to solve an augmented matrix.

Writing and using a linear system You have $10,000 to invest. You want to invest the money in a stock mutual fund, a bond mutual fund, and a money market fund. The expected annual returns for these funds are 10% for stock, 7% for bond and 5% for money market. You want your investment to obtain an overall annual return of 8%. A financial planner recommends that you invest the same amount in stocks as in bonds and the money market combined. How much should you invest in each fund? Pg. 232 in your book

Ex. 3: Write an augmented matrix to represent the system shown. System of Equations x + 2y +3z = -4 y – 2z = 8 z = -3 System of Equations An augmented matrix that represents the system Use the rref( key on your calculator, 12th item under Matrix, MATH Solution is 1,2, -3

Warm up You have $20,000 to invest in three types of stocks. You expect the annual returns on Stock X, Stock Y, and Stock Z to be 12%, 10% and 6% respectively. You want the combined investment in Stock Y and Stock Z to be three times the amount invested in Stock X. You want your overall annual return to be 9%. Write a system of equations and use matrices to determine how much you should invest in each type of stock. $5000, $7500, $7500

Ex. 2: Write an augmented matrix to represent the system shown. System of Equations x - 5y = 15 3x +3y = 3 System of Equations An augmented matrix that represents the system