7.2
Matrix Algebra
Copyright © 2011 Pearson, Inc.
What you’ll learn about
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Matrices
Matrix Addition and Subtraction
Matrix Multiplication
Identity and Inverse Matrices
Determinant of a Square Matrix
Applications
… and why
Matrix 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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Slide 7.2 - 2
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 in 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?
é1 4 5 ù
ê
ú
ë3 5 6 û
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Example Determining the Order of a
Matrix
What is the order of the following matrix?
é1 4 5 ù
ê
ú
ë3 5 6 û
The matrix has 2 rows and 3 columns
so it has order 2 ´ 3.
Copyright © 2011 Pearson, Inc.
Slide 7.2 - 6
Matrix Addition and Matrix
Subtraction
Let A = éë aij ùû and B = éë bij ùû be matrices of order m ´ n.
1. The sum A + B is the m ´ n matrix
A + B = éë aij + bij ùû .
2. The difference A - B is the m ´ n matrix
A - B = éë aij - bij ùû .
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Example Matrix Addition
é 1 2 3ù é 2 3 4 ù
ê
ú+ê
ú
ë4 5 6û ë5 6 7 û
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Example Matrix Addition
é 1 2 3ù é 2 3 4 ù
ê
ú+ê
ú
ë4 5 6û ë5 6 7 û
é1+ 2 2 + 3 3 + 4 ù é 3 5 7 ù
A+ B = ê
ú=ê
ú
ë 4 + 5 5 + 6 6 + 7 û ë9 11 13û
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Example Using Scalar Multiplication
é 1 2 3ù
3ê
ú
ë4 5 6û
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Slide 7.2 - 10
Example Using Scalar Multiplication
é 1 2 3ù
3ê
ú
ë4 5 6û
é 3×1 3× 2 3× 3ù é 3 6 9 ù
=ê
ú=ê
ú
ë3× 4 3×5 3× 6 û ë12 15 18 û
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Slide 7.2 - 11
The Zero Matrix
The m ´ n matrix O = [0] consisting entirely of
zeros is the zero matrix.
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Additive Inverse
Let A = éë aij ùû be any m ´ n matrix.
The m ´ n matrix B = éë aij ùû consisting of the additive
inverses of the entries of A is the additive inverse of A
because A + - B = O.
Copyright © 2011 Pearson, Inc.
Slide 7.2 - 13
Matrix Multiplication
Let A = éë aij ùû be any m ´ r matrix and B = éë bij ùû
be any r ´ n matrix.
The product AB = éë cij ùû is the m ´ n matrix where
cij = ai1b1 j +ai2 b2 j + ... + air brj .
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Slide 7.2 - 14
Example Matrix Multiplication
Find the product AB if possible.
é1 2 3 ù
A= ê
ú
ë0 1 -1û
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é1 0 ù
ê
ú
and B = ê 2 1 ú
êë 0 -1úû
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Example Matrix Multiplication
é1 2 3 ù
A= ê
ú
ë0 1 -1û
é1 0 ù
ê
ú
and B = ê 2 1 ú
êë 0 -1úû
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 = éë cij ùû is a 2 ´ 2 matrix where
é1ù
ê ú
é
ù
c11 = ë1 2 3û ê 2 ú = 1×1+ 2 × 2 + 3× 0 = 5,
êë 0 úû
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Slide 7.2 - 16
Example Matrix Multiplication
é1 2 3 ù
A= ê
ú
ë0 1 -1û
é1 0 ù
ê
ú
and B = ê 2 1 ú
êë 0 -1úû
é0ù
ê ú
c12 = éë1 2 3ùû ê 1 ú = 1× 0 + 2 ×1+ 3× -1 = -1,
êë -1úû
é1ù
ê ú
c21 = éë0 1 -1ùû ê 2 ú = 0 ×1+ 1× 2 + -1× 0 = 2,
êë 0 úû
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Slide 7.2 - 17
Example Matrix Multiplication
é1 2 3 ù
A= ê
ú
ë0 1 -1û
é1 0 ù
ê
ú
and B = ê 2 1 ú
êë 0 -1úû
é0ù
ê ú
c22 = éë0 1 -1ùû ê 1 ú = 0 × 0 + 1×1+ -1× -1 = 2.
êë -1úû
é 5 -1ù
Thus AB = ê
ú.
ë2 2 û
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Slide 7.2 - 18
Identity Matrix
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Inverse of a Square Matrix
Let A = éë aij ùû be an n ´ n matrix.
If there is a matrix B such that
AB = BA = I n ,
then B is the inverse of A. We write B = A-1 .
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Slide 7.2 - 20
Inverse of a 2 × 2 Matrix
If ad - bc ¹ 0, then
-1
éa b ù
1 é d -b ù
ê
ú = ad - bc ê
ú.
ëc d û
ë -c a û
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Slide 7.2 - 21
Determinant of a Square Matrix
Let A = éë aij ùû 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 + ai2 Ai2 + ... + ain Ain .
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Slide 7.2 - 22
Inverses of n  n Matrices
An n  n matrix A has an inverse if and only if
det A ≠ 0.
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Slide 7.2 - 23
Example Finding Inverse Matrices
Determine whether the matrix has an inverse.
If so, find its inverse matrix.
é5 1ù
A= ê
ú
ë8 3û
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Slide 7.2 - 24
Example Finding Inverse Matrices
é5 1ù
A= ê
ú
ë8 3û
Since det A = ad - bc = 5× 3 - 1×8 = 7 ¹ 0,
we conclude that A has an inverse.
1 é d -b ù 1 é 3 -1ù
Use the formula A =
= ê
ê
ú
ad - bc ë -c a û 7 ë -8 5 úû
é 3
1ù
ê 7 -7ú
=ê
ú.
ê- 8 5 ú
êë 7 7 úû
-1
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Slide 7.2 - 25
Example Finding Inverse Matrices
é5 1ù
A= ê
ú
ë8 3û
Check:
é 3
é 15 8
1ù
3 3 ù
- ú
- ú
ê
ê
é5 1ù
7
7
7 7
7 7
-1
A A= ê
=
úê
ê
ú
ú
ê - 8 5 ú ë8 3û ê - 40 + 40 - 8 + 15 ú
êë 7 7 úû
êë 7
7
7 7 úû
é1 0 ù
=ê
ú = I2
ë0 1 û
Similarly, AA-1 = I 2 .
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Slide 7.2 - 26
Properties of Matrices
Let A, B, and C be matrices whose orders are such that
the following sums, differences, and products are
defined.
1. Commutative property
Addition: A + B = B + A
Multiplication: Does not hold in general
2. Associative property
Addition: (A + B) + C = A + (B + C)
Multiplication: (AB)C = A(BC)
3. Identity property
Addition: A + O = A
Multiplication: A·In = In·A = A
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Slide 7.2 - 27
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 property
Addition: A + (–A) = O
Multiplication: AA–1 = A–1A = In |A| ≠ 0
5. Distributive property
Multiplication over addition:
A(B + C) = AB + AC (A + B)C = AC + BC
Multiplication over subtraction:
A(B – C) = AB – AC (A – B)C = AC – BC
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Slide 7.2 - 28
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 x cos y + sin y cos x
5. cos(x + y) cos x cos y - sin x sin y
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