Answer to Question #117797 in Algebra for mike

Question

size
Mon
Tues
Wed
Thu
Fri
Sat
Sun
Small
28
35
28
36
37
34
45
Medium
36
37
47
32
36
38
47
large
41
42
40
35
51
56
61
Family
30
33
37
33
78
76
83
a) Write the information in the table above as a 4 × 7 matrix, P.

b) Write down a suitable matrix, A, that when P is multiplied by A the resulting matrix will give the
number of each size of pizza sold in the week.

c) Calculate the matrix PA.



d) Write down a suitable matrix, B, that when B is multiplied by P the resulting matrix will give the
number of pizzas sold on each day of the week.

e) Calculate the matrix BP.

Answer 1

Solution.

$P=\begin{pmatrix} 28& 35& 28 &36& 37& 34 &45 \\ 36& 37 &47 &32 &36& 38 &47\\ 41& 42 &40& 35& 51& 56& 61\\ 30& 33 &39& 33& 78& 76 &83 \end{pmatrix}; A=\begin{pmatrix} 1 \\ 1\\ 1\\ 1\\1\\1\\1 \end{pmatrix};$

The product of matrices:

$PA=\begin{pmatrix} 28& 35& 28 &36& 37& 34 &45 \\ 36& 37 &47 &32 &36& 38 &47\\ 41& 42 &40& 35& 51& 56& 61\\ 30& 33 &39& 33& 78& 76 &83 \end{pmatrix}\begin{pmatrix} 1 \\ 1\\ 1\\1\\1\\1\\ 1 \end{pmatrix}=$

$=\begin{pmatrix} 243 \\ 273\\ 326\\ 372 \end{pmatrix},$ where

$A_{ 1,1} ​ =28⋅1+35⋅1+28⋅1+36⋅1+37⋅1+34⋅1+45⋅1=243;$

$A_{2,1}​=36⋅1+37⋅1+47⋅1+32⋅1+36⋅1+38⋅1+47⋅1=273;$

$A_{3,1}=41⋅1+42⋅1+40⋅1+35⋅1+51⋅1+56⋅1+61⋅1=326;$

$A_{4,1}=30⋅1+33⋅1+39⋅1+33⋅1+78⋅1+76⋅1+83⋅1=372;$

$B=\begin{pmatrix} 1& 1& 1 &1& \end{pmatrix};$

$BP=\begin{pmatrix} 1& 1& 1 &1& \end{pmatrix}\begin{pmatrix} 28& 35& 28 &36& 37& 34 &45 \\ 36& 37 &47 &32 &36& 38 &47\\ 41& 42 &40& 35& 51& 56& 61\\ 30& 33 &39& 33& 78& 76 &83 \end{pmatrix}=$ $=\begin{pmatrix} 135&147 & 154&136&202&204& 236 \end{pmatrix},$ where

$A_{1,1}=1\sdot28+1\sdot36+1\sdot41+1\sdot30=135;$

$A_{1,2}=1\sdot35+1\sdot37+1\sdot42+1\sdot33=147;$

$A_{1,3}=1\sdot28+1\sdot47+1\sdot40+1\sdot39=154;$

$A_{1,4}=1\sdot36+1\sdot32+1\sdot35+1\sdot33=136;$

$A_{1,5}=1\sdot37+1\sdot36+1\sdot51+1\sdot78=202;$

$A_{1,6}=1\sdot34+1\sdot38+1\sdot56+1\sdot76=204;$

$A_{1,7}=1\sdot45+1\sdot47+1\sdot61+1\sdot83=236;$

Answer: $PA=\begin{pmatrix} 243 \\ 273\\ 326\\ 372 \end{pmatrix};$

$BP=\begin{pmatrix} 135&147 & 154&136&202&204& 236 \end{pmatrix}.$