4.2 An Introduction to Matrices

Algebra 2

Learning Targets

I can create a matrix and name it using its dimensions

I can perform scalar multiplication on a matrix

I can add and subtract matrices I can find unknown values in equal

matrices.

What is a matrix?

A system of rows and columns A problem-solving tool that organizes numbers

or data so that each position in the matrix has a purpose.

The dimensions of a matrix is read as the number of rows by the number of columns.

A matrix with 2 rows and 3 columns is a

2 X 3, or a 2 by 3.

Connection

In Algebra, a matrix is not expressed as a table, but as an array of values. Each value is called an element of the matrix. Suppose we want to write the coordinates of the vertices of the ABC as a matrix.

Connection Suppose we want to

write the coordinates of the vertices of the ABC as a matrix. Let row 1 be the x-coordinates. Let row 2 be the y-coordinates. Each column represents the coordinates of vertices A, B, and C.

412

123

3 columns

2 rows

4

2

-2

-4

-5 5

C

A

B

How is it named?

Usually named using an uppercase letter. We might call the matrix for ABC matrix T to stand for triangle. A matrix can also be named by using the matrix dimensions with the letter name. The dimensions tell how many rows and columns there are in the matrix. The matrix on the preceding slide would be named T2x3 since it has two rows and 3 columns. Note: Please realize that T2x3 does not have the same dimensions as T3x2 .

What if they don’t have the same number of columns or rows? What if they do?

Certain matrices have special names. A matrix that has only one row is called a

row matrix. A matrix that has only one column is called

a column matrix. A matrix that has the same number of

rows and columns is called a square matrix.

Yes, but are they equal?

Two matrices are considered equal if they have the same dimensions and each element of one matrix is equal to the corresponding element of the other matrix.

Definition of Equal Matrices

Two matrices are equal if and only if they have the same dimensions and their corresponding elements are equal.

Ex. 1: Solve for x and y

x

y

y

x

26

8626

Since the matrices are equal, the corresponding elements are equal. When you write the sentences that show this equality, two linear equations are formed.

6x = 62 + 8y

Y = 6 – 2x

Ex. 1: Solve for x and yThe second equation gives you a value for y that can be substituted into the first equation. Then you can find a value for x.

6x = 62 + 8y

y = 6 – 2x

6x = 62 + 8y

6x = 62 + 8(6 – 2x)

6x = 62 + 48 – 16x

22x = 110

x = 5

Write down the first equation.

Substitute 6 – 2x for y.

Use the distributive property.

Add 16x to both sides and combine like terms.

Divide by 22.

Ex. 1: Solve for x and yTo find a value for y, you can substitute 5 into either equation.

y = 6 – 2x

y = 6 – 2(5)

y = 6 – 10

y = -4

Check your solutions by substituting values into the equation YOU DID NOT USE to find y.

6x = 62 + 8y

6(5) = 62 + 8(-4)

30 = 62 – 32 or 30✔

Write down the second equation.

Substitute 5 for x.

Use the distributive property.

Simplify.

Scalar Multiplication

You can multiply any matrix by a constant. This is called scalar multiplication. When scalar multiplication is performed, each element is multiplied by that constant, and a new matrix is formed.

kfkekd

kckbka

fed

cbak

Scalar Multiplication

Scalar multiplication of matrices can be used to find the coordinates of the vertices of a geometric figure that is enlarged or reduced.

When the size of a figure changes, the measures of its sides change in the same proportion.

For example, if a figure triples in perimeter, its sides triple in length.

Ex. 2: Enlarge ABC, with vertices A(3, 2), B(-2, 1), C(1, -4), so that its perimeter is twice the perimeter of the original figure. Graph ABC, then

multiply the coordinates by 2.

824

246

412

1232

6

4

2

-2

-4

-6

-8

-10 -5 5 10

C'

B'

A'

C

B

A

The coordinates of the vertices of A’B’C’ are (6, 4), (-4, 2), (2, -8). The two triangles are similar. The perimeter of A’B’C’ is twice the perimeter of ABC

Addition of Matrices Matrices can also be added. In order to

add tow matrices, they must have the same dimensions.

If A and B are two m x n matrices, then A + B is an m x n matrix where each element is the sum of the corresponding elements of A and B.

riqhpg

ofnemd

lckbja

rqp

onm

lkj

ihg

fed

cba

Addition of Matrices

When a figure is moved from one location to another on the coordinate plane without changing its orientation, size, or shape, a translation occurs. You can use matrix addition to find the coordinates of the translated figures.

Ex. 3: Find the coordinates of quadrilateral QUAD if the figure is moved 5 units to the right and 1 unit down.

8

6

4

2

-2

-4

-6

-8

-10 -5 5 10

UQ

D

A

1. Write the coordinates of quadrilateral QUAD in the form of a matrix.

1422

2312

2. To translate the quadrilateral 5 units to the right means that each x-coordinate increases by 5. Translating the figure 1 unit down decreases each y-coordinate by 1.

Ex. 3: Find the coordinates of quadrilateral QUAD if the figure is moved 5 units to the right and 1 unit down.

1111

5555

The matrix, called a translation matrix, that increases each x-value by five and decreases each y-value by 1 is:

3. To find the coordinates of the translated Q’U’A’D’, add the two matrices.

0333

3863

1111

5555

1422

2312

Ex. 3 continued:

4. Now graph the coordinates of quadrilateral Q’U’A’D’ to check the accuracy of your coordinates. The two quadrilaterals have the same size and shape. Quadrilateral Q’U’A’D’ is QUAD moved right 5 units and down 1 unit.

6

4

2

-2

-4

-6

-8

-10

-5 5 10 15D'

A'

U'Q'

UQ

D

A