Transcript Lesson 4.1 Operation of Matrices

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
4

Suppose we want to
write the
coordinates of the
vertices of the
ABC as a matrix.
Let row 1 be the xcoordinates. Let
row 2 be the ycoordinates. Each
column represents
the coordinates of
vertices A, B, and C.
A
2
B
-5
5
-2
-4
C
3 columns
2 rows
3  2 1
 2 1  4


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
6 x  62  8 y 
 y   6  2 x 
  

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 y
The 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
Write down the first equation.
6x = 62 + 8(6 – 2x)
Substitute 6 – 2x for y.
6x = 62 + 48 – 16x
Use the distributive property.
22x = 110
x=5
Add 16x to both sides and combine like terms.
Divide by 22.
Ex. 1: Solve for x and y
To find a value for y, you can substitute 5 into either equation.
y = 6 – 2x
Write down the second equation.
y = 6 – 2(5) Substitute 5 for x.
y = 6 – 10
y = -4
Use the distributive property.
Simplify.
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✔
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.
a b c  ka kb kc 
k



d e f  kd ke kf 
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.
3  2 1 6  4 2
2



 2 1  4  4 2  8 
6
A'
4
B'
A
2
B
-10
-5
5
10
-2
-4
C
-6
-8
C'
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.

a b c   j k l   a  j b  k c  l 
 d e f   m n o   d  m e  n f  o 

 
 

 g h i   p q r   g  p h  q i  r 
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.
1. Write the coordinates of
quadrilateral QUAD in the
form of a matrix.
8
6
4
A
2
D
-10
-5
5
-2
Q
U
-4
-6
-8
10
 2 1 3  2 
 2  2 4 1


2. To translate the
quadrilateral 5 units to the
right means that each xcoordinate increases by 5.
Translating the figure 1 unit
down decreases each ycoordinate by 1.
Ex. 3: Find the coordinates of quadrilateral QUAD
if the figure is moved 5 units to the right and 1 unit
down.
The matrix, called a translation matrix, that increases each x-value by five
and decreases each y-value by 1 is:
5 5 5 5
1 1 1 1 


3. To find the coordinates of the translated Q’U’A’D’, add the two matrices.
 2 1 3  2  5 5 5 5
 3 6 8 3 
 2  2 4 1   1  1  1  1   3  3 3 0

 
 

Ex. 3 continued:
6
4
A
A'
2
D
D'
-5
5
10
-2
Q
U
Q'
-4
-6
-8
-10
U'
15
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.