Autar Kaw Benjamin Rigsby - MATH FOR...
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Autar Kaw Benjamin Rigsby http://nm.MathForCollege.com Transforming Numerical Methods Education for STEM Undergraduates
Transcript of Autar Kaw Benjamin Rigsby - MATH FOR...
Autar KawBenjamin Rigsby
http://nm.MathForCollege.comTransforming Numerical Methods Education for STEM Undergraduates
1. add, subtract, and multiply matrices, and
2. apply rules of binary operations on matrices.
Two matrices and can be added only if they are the same size. The addition is then shown as
][][][ BAC +=
ijijij bac +=
where
Add the following two matrices.
.
=
721325
][A
−=
1953276
][B
][][][ BAC +=
−+
=
1953276
721325
+++−++
=1975231237265
=
26741911
×
Blowout r’us store has two store locations and , and their sales of tires are given by make (in rows) and quarters (in columns) as shown below.
×
=
2771662515105232025
][A
=
2071421156304520
][B
where the rows represent the sale of Tirestone, Michigan and Copper tires respectively and the columns represent the quarter number: 1, 2, 3 and 4. What are the total tire sales for the two locations by make and quarter?
ABCDG d trapezoiofArea =
][][][ BAC +=
+
=
2771662515105232025
2071421156304520
( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )
++++++++++++
=20277711646212515156103502435202025
So if one wants to know the total number of Copper tires sold in quarter 4 at the two locations, we would look at Row 3 – Column 4 to give .4734 =c
=
471417104630168272545
The answer then is,
Two matrices ][A and ][B can be subtracted only if they are the same size. The subtraction is then given by
][][][ BAD −=
Where
ijijij bad −=
Subtract matrix ][B from matrix .][A
=
721325
][A
−=
1953276
][B
][][][ BAD −=
−−
=
1953276
721325
−−−−−−−
=)197()52()31())2(3()72()65(
−−−
−−=
1232551
×
Blowout r’us store has two store locations and , and their sales of tires are given by make (in rows) and quarters (in columns) as shown below.
×
=
2771662515105232025
][A
=
2071421156304520
][B
where the rows represent the sale of Tirestone, Michigan and Copper tires respectively and the columns represent the quarter number: 1, 2, 3 and 4. What are the total tire sales for the two locations by make and quarter?
][][][ BAD −=
+
=
2771662515105232025
2071421156304520
−−−−−−−−−−−−
=20277711646212515156103502435202025
So if you want to know how many more Copper tires were sold in quarter 4 in store A than store B, . Note that implies that store A sold 1 less in Michigan tire than store B in quarter 3.
734 =d 113 −=d
−=
70152404221155
The answer then is,
Two matrices [A] and [B] can be multiplied only if the number of columns of [A] is equal to the number of rows of [B] to give
nppmnm BAC ××× = ][][][
If ][A is a pm× matrix and ][B is a np× matrix, the resulting matrix ][C
is a nm× matrix.
So how does one calculate the elements of ][C matrix?
∑=
=p
kkjikij bac
1
pjipjiji bababa +++= 2211
for each mi ,,2 ,1 = and nj ,,2 ,1 =
To put it in simpler terms, the thi row and thj column of the ][C matrix in
]][[][ BAC = is calculated by multiplying thethi row of ][A by the thj
column of . ][B
Given the following two matrices,
=
721325
][A
−−−
=1098523
][B
Find their product,
[ ] [ ][ ]BAC =
12c be found by multiplying the first row of ][A by the second column of ,][B
[ ]
−−−
=1082
32512c
)10)(3()8)(2()2)(5( −+−+−=
56−=
Similarly, one can find the other elements of ][C to give
−−
=88765652
][C
×
Blowout r’us store has two store locations and , and their sales of tires are given by make (in rows) and quarters (in columns) as shown below.
×
=
2771662515105232025
][A
=
2071421156304520
][B
where the rows represent the sale of Tirestone, Michigan and Copper tires respectively and the columns represent the quarter number: 1, 2, 3 and 4.
Find the per quarter sales of store A if the following are the prices of eachtire.
Tirestone = $33.25Michigan = $40.19Copper = $25.03
The answer is given by multiplying the price matrix by the quantity of sales of store
A. The price matrix is
A
A
. The price matrix is .[ ]03.2519.4025.33
Therefore, the per quarter sales of store A dollars is given by the four columns of the row vector
[ ] [ ]06.174781.87738.146738.1182=C
×Remember since we are multiplying a 1 3 matrix by a 3
resulting matrix is a 1 4 matrix.
4 matrix, the××
If ][A is a nn× matrix and k is a real number, then the scalar product of
k and ][A is another nn× matrix , where . ][Bijij akb =
Given the matrix,
=
615231.2
][A
][2 AFind
=
615231.2
2][2 A
××××××
=62125222321.22
=
12210462.4
The solution to the product of a scalar and a matrix by the following method,
][],.....,[],[ 21 pAAA pkkk ,.....,, 21
][........][][ 2211 pp AkAkAk +++
][],.....,[],[A 21 pAA
If are matrices of the same size and are scalars,then is called a linear combination of
=
=
=
65.3322.20
][,615231.2
][,123265
][ 321 AAA
If
then find
][5.0][2][ 321 AAA −+
][5.0][2][ 321 AAA −+
−
+
=
65.3322.20
5.0615231.2
2123265
−
+
=
375.15.111.10
12210462.4
123265
=
1025.25.1159.102.9
Commutative law of addition
If [A] and [B] are m×n matrices, then [A]+[B]=[B]+[A]
Associative law of addition
If [A], [B] and [C] are m×n, n×p, and p×r size matrices, respectively, then[A]+([B]+[C])=([A] +[B])+[C]
Associative law of multiplication
If [A], [B] and [C] are all m×n, n×p and p×r size matrices, respectively, then[A]([B][C])=([A][B])[C]
and the resulting matrix size on both sides of the equation is m×r.
If [A] and [B] are m×n matrices, and [C] and [D] are n×p size matrices[A]([C]+[D])=[A][C]+[A][D]([A]+[B])[C]=[A][C]+[B][C]
and the resulting matrix size on both sides of the equation is m×r.
Distributive Law
Illustrate the associative law of multiplication of matrices using
=
=
=
5312
][,6952
][,205321
][ CBA
=]][[ CB
=
5312
6952
=
39362719
=
39362719
205321
])][]([[ CBA
=
787227623710591
=
6952
205321
]][[ BA
=
121845511720
=
5312
121845511720
]])[][([ CBA
=
787227623710591
If [A][B] exists, number of columns of [A] has to be same as the number of rows
of [B] and if [B][A] exists, number of columns of [B] has to be same as the
number of rows of [A].
Now for [A][B]=[B][A], the resulting matrix from [A][B] and [B][A] has to be of
the same size. This is only possible if [A] and [B] are square and are of the same
size. Even then in general [A][B]≠[B][A].
Determine if
[A][B]=[B][A]
for the following matrices
−=
=
5123
][,5236
][ BA
]][[ BA
−
=
5123
5236
−−
=2912715
]][[ AB
−=
5236
5123
−=
2816114
]][[]][[ ABBA ≠Therefore
