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Lesson7.2
Matrix Algebra
Precalculus
The points and are reflected across the given line.
Find the coordinates of the reflected points.
1. The -axis
2. The line
3. The line
Expand the ex
(
p
(a) (1, 3
ression,
4. sin( )
5. co
b )) ( ,
s
) x
x
y
x
y x
y
x y
( )x y
Quick Review
( ) (1,3)a ( ) ( , )b x y
( ) ( 3,1)a ( ) ( , )b y x
( ) (3, 1)a ( ) ( , )b y x
sin cos sin cosx y y x
cos cos sin sinx y x y
What you’ll learn about
MatricesMatrix Addition and SubtractionMatrix MultiplicationIdentity and Inverse MatricesDeterminant of a Square MatrixApplications
… and whyMatrix algebra provides a powerful technique to manipulate
large data sets and solve the related problems that are modeled by the matrices.
Matrix
Let and be positive integers.
An (read " by matrix") is a rectangular
array of rows and columns of real numbers.
m n
m n
m n
matrixm×n
11 12 1
21 22 2
1 2
n
n
m m mn
a a a
a a a
a a a
We also use the shorthand notation for this matrix.ija
Each element, or entry, aij, of the matrix uses double subscript notation. The row subscript is the first subscript i, and the column subscript is j. The element aij is the ith row and the jth column. In general, the order of an m × n matrix is m×n.
Matrix Vocabulary
Example Determining the Order of a Matrix
What is the order of the following matrix?
1 4 5
3 5 6
The matrix has 2 rows and 3 columns so
ordeit h r as 2 .3
Matrix Addition and Matrix Subtraction
Let and be matrices of order .ij ij
A a B b m n
1. The is the matrix
.ij ij
m n
A B a b
sum +A B
2. The is the matrix
.ij ij
m n
A B a b
difference A - B
Example Matrix Addition
1 2 3 2 3 4
4 5 6 5 6 7
2 3 4
5
1 2 3
4 65 76
2 3 4
5
1 2 3
4 5 6 6 7
3 5 7
9 11 13
Example Using Scalar Multiplication
1 2 33
4 5 6
3 6 9
12 15 18
31 2 3
4
3 3 3
3 53 63
The Zero Matrix
The matrix 0 [0] consisting
entirely of zeros is the .
m n zero matrix
0 0 0
0 0 0
Example:
Additive Inverse
Let be any matrix.
The matrix consisting of
the additive inverses of the entries of
is the because
0.
ij
ij
A a m n
m n B a
A
A B
additive inverse of A
Example Using Additive Inverse
Given matrix define below
2
0.5
6
find its additive inverse matrix ?
A
A
B
2 0
0.5 0
6 0
?
?
?
2
0.5
6
B
Matrix Multiplication
1 1 2 2
Let be any matrix and
be any matrix.
The product is the matrix
where + ... .i j i j
ij
ij
ij
ij ir rj
A a m r
B b r n
AB c m n
c a b a b a b
Example Matrix Multiplication
? ?
? ?AB
3 is 2A 2 is 3B
is 2 2 AB
1 1 2 3
?
0
?
2 ?
1 0 2 1 3
?
15
?
0 1 2
5 1
1 0 ?1
5 1
2 0 0 1 1 1 1
5 1
2 2
Find the product if possible.
1 01 2 3
and 2 1 0 1 1
0 1
AB
A B
Identity Matrix
The matrix with 1's on the main diagonal
and 0's elsewhere is the
.
1 0 0 0
0 1 0 0
0 0 1 0
0
0 0 0 0 1
n
n
n n I
I
identity matrix
of order
n n
Inverse of a Square Matrix
-1
Let be an matrix.
If there is a matrix such that
,
then is the of .
We write .
ij
n
A a n n
B
AB BA I
B A
B A
inverse
Example Inverse of a Square Matrices
Determine whether the matrices are inverses.
5 3 2 3,
3 2 3 5A B
1 0
0 1
10 9 15 15
6 6 9 10 AB
10 9 6 6
15 15
9 10BA
1 0
0 1
YES
Inverse of a 2 × 2 Matrix
1
If 0,
1then .
ad bc
a b d b
c d c aad bc
The number is the determinant
of the 2 2 matrix .
ad bc
a bA
c d
Determinant of a Square Matrix
Let be a matrix of order ( 2).
The determinant of , denoted by det or | | ,
is the sum of the entries in any row or any column
multiplied by their respective cofactors.
For example, expa
ijA a n n n
A A A
1 1 2 2
nding by the th row gives
det | | ... .i i i i in in
i
A A a A a A a A
Refer to text pg 583
Inverses of n × n Matrices
An n × n matrix A has an inverse if and only if det A ≠ 0.
Example Finding Inverse Matrices
1 3Find the inverse matrix if possible.
2 5A
1 1Use the formula
d bA
c aad bc
Since det 1 5 2 3 1 0,
must have an inverse.
A ad bc
A
5 3
2 1
15 3
1
2 11A
Properties of MatricesLet A, B, and C be matrices whose orders are such thatthe following sums, differences, and products are defined.
1. Commutative propertyAddition: A + B = B + AMultiplication: Does not hold in general
2. Associative propertyAddition: (A + B) + C = A + (B + C)Multiplication: (AB)C = A(BC)
3. Identity propertyAddition: A + 0 = AMultiplication: A·In = In·A = A
Properties of Matrices
Let A, B, and C be matrices whose orders are such thatthe following sums, differences, and products are defined.
4. Inverse propertyAddition: A + (-A) = 0Multiplication: AA-1 = A-1A = In |A|≠0
5. Distributive propertyMultiplication over addition: A(B + C) = AB + AC
(A + B)C = AC + BCMultiplication over subtraction: A(B - C) = AB - AC
(A - B)C = AC - BC
Homework:
Text pg588/589 Exercises #2, 4, 14, 20, 24, and 34