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

MatrixA rectangular arrangement of numbers in

rows and columns

Dimensions of a Matrix: # rows by # columns

2 X 3 (read 2 by 3)

701

436

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.

3241

1071

265305712

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

441

217

425.

5.7

5321

3810

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.

891

340

53

025933

8431

79

Scalar Multiplication: multiply each entry of the matrix by the scalar

1.

2.

4832

6

418963

923374

1

Solve for x and y:

71959

2410

52

3y

x

71959

2410

31563y

x

71959

231953y

x

3x = -9, x = -33y-2 =7, y = 3

Properties of Matrix Operations: A, B and C are matrices, c is a scalar

1. Associative Property (regroup)

2. Commutative Property (change order

3. Distributive Property of Addition

4. 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.

51.89291.48616.98429.51832.115015.612

1 car 2 carsComp.

Coll.

basic

13.89505.49072.98739.52084.115528.616

This year (A) Next year (B)

B – A will give the change of:

62.214.356.31.252.513.4

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.

831020211320

312

1 x 3 3 x 3

Must be the same

Dimensions of product matrix

DeterminantsThe determinant of a matrix is the

difference in the cross products

det A or lAl

bcdadcba

Find the determinant of a 2 x 2 matrix

14410)14()52(5412

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.

heb

gda

ihgfedcba

ihgfedcba

Using Diagonals• STEP 2: Draw a diagonal from each element in

the top row diagonally downward. Find the product of the numbers on each diagonal.

heb

gda

ihgfedcba

ihgfedcba

aei bfg cdh

Using Diagonals• STEP 3: Then draw a diagonal from each

element in the bottom row diagonally upward. Find the product of the numbers on each .

heb

gda

ihgfedcba

ihgfedcba

idbhfagec

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.

124

331

213523041

213523041

First, rewrite the first two columns along side the determinant.

Ex. 2: Evaluate using diagonals.

124

331

213523041

213523041

Next, find the values using the diagonals.

4 60 0

0 -5 24

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:

,111

21

fedcba

A 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.

,111

21

fedcba

A a = -4, b = -1, c = 3, d = 2, e = 4, f = 6

621

434

164123114

21

A

-8 -4 18

8 -24-3

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.

1) Inverse Matrices and Systems of Equations

For a We can write a System of Equations Matrix Equation

145352

yxyx

145

5321yx

1) Inverse Matrices and Systems of Equations

Example 1:Write the system as a matrix equation

Matrix Equation

621132

yxyx

611

2132yx

Coefficient matrix

Constant matrix

Variable matrix

1) Inverse Matrices and Systems of Equations

BAX 1

When rearranging, take the inverse of A

BAX

1) Inverse Matrices and Systems of Equations

Example 3:Solve the system

Step 3: Solve for the variable matrix

14

611

2132

1

1

yx

yx

BAyx

BAX

The solution to the system is (4, 1).

621132

yxyx

1) Inverse Matrices and Systems of Equations

Example 2:

A BX

850

212121111

zyx

822520

zyxzyxzyx

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.

2041026

xyx

System of Equations -6x - 2y = 10

4x = -20

System of Equations

2010

|04|26

Use the rref key under the matrix math menu to solve an augmented matrix.

Ex. 2: Write an augmented matrix to represent the system shown.

333155

yxyx

System of Equations x - 5y = 15

3x +3y = 3

System of Equations

315|33|51

An augmented matrix that represents the system

Ex. 3: Write an augmented matrix to represent the system shown.

382

432

zzy

zyx

System of Equations x + 2y +3z = -4

y – 2z = 8

z = -3

System of Equations

310082104321

An augmented matrix that represents the system