autar kaw benjamin rigsby - math for...
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Autar KawBenjamin Rigsby
http://nm.MathForCollege.comTransforming Numerical Methods Education for STEM Undergraduates
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1. add, subtract, and multiply matrices, and
2. apply rules of binary operations on matrices.
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Two matrices and can be added only if they are the same size. The addition is then shown as
][][][ BAC +=
ijijij bac +=
where
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Add the following two matrices.
.
=
721325
][A
−=
1953276
][B
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][][][ BAC +=
−+
=
1953276
721325
+++−++
=1975231237265
=
26741911
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×
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?
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ABCDG d trapezoiofArea =
][][][ BAC +=
+
=
2771662515105232025
2071421156304520
( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )
++++++++++++
=20277711646212515156103502435202025
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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,
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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 −=
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Subtract matrix ][B from matrix .][A
=
721325
][A
−=
1953276
][B
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][][][ BAD −=
−−
=
1953276
721325
−−−−−−−
=)197()52()31())2(3()72()65(
−−−
−−=
1232551
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×
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?
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][][][ BAD −=
+
=
2771662515105232025
2071421156304520
−−−−−−−−−−−−
=20277711646212515156103502435202025
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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,
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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.
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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
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Given the following two matrices,
=
721325
][A
−−−
=1098523
][B
Find their product,
[ ] [ ][ ]BAC =
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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−=
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Similarly, one can find the other elements of ][C to give
−−
=88765652
][C
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×
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.
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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
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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××
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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 =
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Given the matrix,
=
615231.2
][A
][2 AFind
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=
615231.2
2][2 A
××××××
=62125222321.22
=
12210462.4
The solution to the product of a scalar and a matrix by the following method,
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][],.....,[],[ 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
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=
=
=
65.3322.20
][,615231.2
][,123265
][ 321 AAA
If
then find
][5.0][2][ 321 AAA −+
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][5.0][2][ 321 AAA −+
−
+
=
65.3322.20
5.0615231.2
2123265
−
+
=
375.15.111.10
12210462.4
123265
=
1025.25.1159.102.9
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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.
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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
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Illustrate the associative law of multiplication of matrices using
=
=
=
5312
][,6952
][,205321
][ CBA
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=]][[ CB
=
5312
6952
=
39362719
=
39362719
205321
])][]([[ CBA
=
787227623710591
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=
6952
205321
]][[ BA
=
121845511720
=
5312
121845511720
]])[][([ CBA
=
787227623710591
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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].
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Determine if
[A][B]=[B][A]
for the following matrices
−=
=
5123
][,5236
][ BA
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]][[ BA
−
=
5123
5236
−−
=2912715
]][[ AB
−=
5236
5123
−=
2816114
]][[]][[ ABBA ≠Therefore