unit 7.2
TRANSCRIPT
Copyright © 2011 Pearson, Inc. Slide 7.2 - 2
What you’ll learn about
Matrices Matrix Addition and Subtraction Matrix Multiplication Identity and Inverse Matrices Determinant of a Square Matrix Applications
… and whyMatrix algebra provides a powerful technique to manipulate large data sets and solve the related problems that are modeled by the matrices.
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Matrix
Let m and n be positive integers. An m n matrix
(read "m by n matrix") is a rectangular array of
m rows and n columns of real numbers.
a11
a12
L a1n
a21
a22
L a2n
M M M
am1
am2
L amn
We also use the shorthand notation aij
for this matrix.
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Matrix Vocabulary
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.
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Example Determining the Order of a Matrix
What is the order of the following matrix?
1 4 5
3 5 6
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Example Determining the Order of a Matrix
The matrix has 2 rows and 3 columns
so it has order 2 3.
What is the order of the following matrix?
1 4 5
3 5 6
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Matrix Addition and Matrix Subtraction
Let A aij
and B bij
be matrices of order m n.
1. The sum A + B is the m n matrix
A B aij b
ij .
2. The difference A B is the m n matrix
A B aij b
ij .
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Example Matrix Addition
A B
2 1 2 3 3 4
4 5 56 6 7
3 5 7
9 11 13
1 2 3
4 5 6
2 3 4
5 6 7
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Example Using Scalar Multiplication
31 32 3334 35 36
3 6 9
12 15 18
3
1 2 3
4 5 6
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The Zero Matrix
The m n matrix 0 [0] consisting entirely of
zeros is the zero matrix.
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Additive Inverse
Let A aij
be any m n matrix.
The m n matrix B aij
consisting of the additive
inverses of the entries of A is the additive inverse of A
because A B 0.
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Matrix Multiplication
Let A aij
be any m r matrix and B bij
be any r n matrix.
The product AB cij
is the m n matrix where
cija
i1b
1 j+a
i2b
2 j ... a
irb
rj.
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Example Matrix Multiplication
Find the product AB if possible.
A 1 2 3
0 1 1
and B
1 0
2 1
0 1
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Example Matrix Multiplication
A 1 2 3
0 1 1
and B
1 0
2 1
0 1
The number of columns of A is 3 and the number of
rows of B is 3, so the product is defined.
The product AB cij
is a 2 2 matrix where
c11
1 2 3
1
2
0
11 22 30 5,
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Example Matrix Multiplication
A 1 2 3
0 1 1
and B
1 0
2 1
0 1
c12
1 2 3
0
1
1
10 213 1 1,
c21
0 1 1
1
2
0
0112 10 2,
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Example Matrix Multiplication
A 1 2 3
0 1 1
and B
1 0
2 1
0 1
c22
0 1 1
0
1
1
00 11 1 12.
Thus AB 5 1
2 2
.
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Identity Matrix
The n n matrix In with 1's on the main diagonal and
0's elsewhere is the identity matrix of order n n.
In
1 0 0 L 0
0 1 0 L 0
0 0 1 L 0
M M M 0
0 0 0 0 1
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Inverse of a Square Matrix
Let A aij
be an n n matrix.
If there is a matrix B such that
AB BA In,
then B is the inverse of A. We write B A 1.
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Inverse of a 2 × 2 Matrix
If ad bc 0, then
a b
c d
1
1
ad bc
d b
c a
.
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Determinant of a Square Matrix
Let A aij
be a matrix of order n n (n 2).
The determinant of A, denoted by det A or | A | ,
is the sum of the entries in any row or any column
multiplied by their respective cofactors. For
example, expanding by the ith row gives
det A | A |ai1
Ai1 a
i2A
i2 ... a
inA
in.
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Inverses of n n Matrices
An n n matrix A has an inverse if and only if
det A ≠ 0.
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Example Finding Inverse Matrices
Determine whether the matrix has an inverse.
If so, find its inverse matrix.
A 5 1
8 3
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Example Finding Inverse Matrices
Since det A ad bc 53 18 7 0,
we conclude that A has an inverse.
Use the formula A 1 1
ad bc
d b
c a
1
7
3 1
8 5
3
7
1
7
8
7
5
7
.
A
5 1
8 3
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Example Finding Inverse Matrices
Check:
A 1 A
3
7
1
7
8
7
5
7
5 1
8 3
3
7
8
7
3
7
3
7
40
7
40
7
8
7
15
7
1 0
0 1
I
2
A
5 1
8 3
Similarly, A 1A I2.
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Properties of Matrices
Let A, B, and C be matrices whose orders are such that the following sums, differences, and products are defined.1. Community propertyAddition: A + B = B + AMultiplication: Does not hold in general2. Associative propertyAddition: (A + B) + C = A + (B + C)Multiplication: (AB)C = A(BC)3. Identity propertyAddition: A + 0 = AMultiplication: A·In = In·A = A
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Properties of Matrices
Let A, B, and C be matrices whose orders are such that the following sums, differences, and products are defined.4. Inverse propertyAddition: A + (-A) = 0Multiplication: AA-1 = A-1A = In |A|≠05. 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
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Quick Review
The points (a) (1, 3) and (b) (x, y) are reflected
across the given line.
Find the coordinates of the reflected points.
1. The x-axis
2. The line y x
3. The line y x
Expand the expression,
4. sin(x y)
5. cos(x y)
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Quick Review Solutions
The points (a) (1, 3) and (b) (x, y) are reflected
across the given line.
Find the coordinates of the reflected points.
1. The x-axis (a) (1,3) (b) (x, y)
2. The line y x (a) ( 3,1) (b) ( y,x)
3. The line y x (a) ( 3, 1) (b) ( y, x)
Expand the expression,
4. sin(x y) sin xcos y sin ycos x
5. cos(x y) cos xcos y sin xsin y