Chapter 2: Systems of linear
equations and inequalities
Section 2-3: Modeling real-world
data with matrices
Objectives
ƒ Model data using matrices
ƒ Add, subtract, and multiply matrices
What is a matrix?
A matrix is a rectangular array of terms
called elements. The elements of a matrix
are arranged in rows and columns and are
usually enclosed by brackets.
A matrix with m rows and n columns is an
m x n matrix. The dimensions of the matrix
are m and n. Important: a 3 x 2 matrix is
NOT the same as a 2 x 3 matrix.
History of Matrices
ƒ Computers use matrices to solve many types of mathematical
problems
ƒ But matrices have been used by man even before computers.
ƒ Around 300 B.C., Babylonian clay tablets were found by
archaeologists with problems that can now be solved using a
system of linear equations. Mathematicians have yet to
determine the exact method the Babylonians used to solve
these problems
ƒ About 100 B.C. in Ancient China, Jiuzhang Suanshu showed
a solution to a problem on a counting board that resembled a
matrix.
This method was later credited to German mathematician
Carl Friedrich Gauss.
Special matrices
ƒ A matrix that has only one row is called a
row matrix.
ƒ A matrix that has only one column is called
a column matrix.
ƒ A square matrix has the same number of
rows as columns.
Equal matrices
ƒ Two matrices are equal if and only if they
have the same dimensions and are
identical, element by element.
Example # 1
Find the values of x and y for which the matrix equation is true.
⎡ y ⎤ ⎡4 x ⎤
⎢
⎥=⎢
⎥
y
−
3
2
x
+
1
⎢⎣
⎥⎦ ⎢⎣
⎥⎦
In order for them to be equal, y=4x and y-3=2x+1.
We have a system of linear equations.
Y=4x
Y=2x+4
So 4x=2x+4
2x+4
X=2
Thus y=8
Addition of matrices
ƒ Important: The sum of two matrices exist
ONLY IF the two matrices have the SAME
dimensions.
ƒ The sum of two m x n matrices is an m x n
matrix in which the elements are the sum
of the corresponding elements of the given
matrices.
Example # 2
Find A + B if A=
-7
4
5 0
3 -1
A+B=
⎡− 1 14⎤
⎢ 13 - 9 ⎥
⎥
⎢
⎢⎣ 1 4 ⎥⎦
and B =
6
10
8 -9
-2 5
Additive Identity Matrix
You know that 0 is the additive identity for
real numbers because a+0=a. Matrices
also have additive identities. For every
matrix A another matrix can be found so
that their sum is A. It is called the zero
matrix and every element is zero. The zero
matrix is the additive identity matrix.
Additive Inverse matrix
ƒ You also know that for any number a, there is a
number –a, called the additive inverse of a, such
that a + -a =0. Matrices also have additive
inverses.
⎡
ƒ If A= ⎢
⎣
a
a
to A
11
21
to
a
a
get
12
22
the
⎤
⎥
⎦
then
zero
the
matrix
matrix wou
that
ld be
must
⎡
⎢
⎣
be
-a
-a
11
21
added
-a
-a
12
22
⎤
⎥
⎦
Subtraction of matrices
The difference A-B of two m X n matrices is equal
to the sum A + (-B) where –B represents the
additive inverse of B.
Find S – T if S= 2 -1 3 and T = -5 -4 1
-4 -2 -8
7 -8 4
S+(-T)=
2 -1 3
-4 -2 -8
+
5 4 -1
-7 8 -4 =
7 3 2
-11 6 -12
Scalar Product
You can multiply a matrix by a number. When you
do this the number is called a scalar.
The product of a scalar K and an m x n matrix A is
an m x n matrix denoted by KA. Each element of
KA equals K times the corresponding element of
A.
⎡20 - 8 ⎤
5
2
⎡
⎤
If A=
4A= ⎢
⎥
⎢
⎥
⎢ 3 8 ⎥ find 4A
⎢⎣- 1 - 9⎥⎦
12
32
⎢
⎥
⎢⎣- 4 - 36⎥⎦
HW #13
ƒ Section 2-3
ƒ Pp 83-86
ƒ #15-37 odds, 56, 60
Product of two matrices
ƒ You can also multiply a matrix by a matrix.
VERY IMPORTANT!!!!
ƒ For matrices A and B you can find AB IF
THE NUMBER OF COLUMNS IN A IS
THE SAME AS THE NUMBER OF ROWS
IN B. IF THEY DO NOT THEN THE
DIMENSIONS OF THE MATRICES DO
NOT ALLOW FOR MULTIPLICATION.
Product of matrices
ƒ So multiplication of matrices is allowed
only if m x n is the first matrix and n x p is
the second matrix. The variables m and p
can have any value, but n must be the
same.
ƒ So you can multiply a 5 x 3 and 3 x 7, but
you can NOT multiply a 3 x 5 and a 3 x 5.
How to multiply matrices
A=
AB=
a1 b1
a2 b2
and B=
x1
x2
y1
y2
a1x1+b1x2
a1y1+b1y2
a2x1+b2x2
a2y1+b2y2
Examples
A=
4
0
3
-1
1
-2
2
0
4
and B=
4
-2
2
3
and C=
1 2 -3
3 1 0
Find AB and BC
AB is impossible because A is a 3 x 3 matrix and B is a 2 x 2
matrix.
BC=
4(1)+2(3)
4(2)+2(1)
4(-3)+2(0)
-2(1)+3(3)
-2(2)+3(1)
-2(-3)+3(0)
10
10
-12
7
-1
6
=
HW #14
ƒ Section 2-3
ƒ Pp83
ƒ #38-42 all