Question

3 persons A, B and C go to stationary shop and if persons A buying 10 dozens notebooks and 4 dozens pen and 5 dozens pencil and person B purchases 8 dozens notebooks and 6 dozens pen and 7 dozens pencil, and person C purchases 10 dozens notebooks and 5 dozens pen and 11 dozens pencil. A notebook cost Rs. 3, a pen cost Rs. 2 and pencil cost Rs. 1, by matrix multiplication to calculate each individual will.

Answer 1

Hint – In this particular question use the concept that in 1 dozen there are 12 items so first calculate the cost of notebooks, pen and pencil per dozen, then write these values in the matrix form and then apply matrix multiplication concept, so use these concepts to reach the solution of the question.

Complete step-by-step answer:
Given data:
A purchase - 10 dozens notebooks, 4 dozens pen and 5 dozens pencil.
B purchase - 8 dozens notebooks, 6 dozens pen and 7 dozens pencil.
C purchase - 10 dozens notebooks, 5 dozens pen and 11 dozens pencil.
Now as we know that in 1 dozen there are 12 items.
Now it is given that the cost of single note book = Rs. 3
Cost of single pen = Rs. 2
And cos of single pencil = Rs. 1
So the cost of one dozen notebooks = cost of 1 pencil multiplied by12.
So the cost of one dozen notebooks = 3(12) = Rs. 36
Now the cost of one dozen pens = cost of 1 pen multiplied by 12.
So the cost of one dozen pens = 2(12) = Rs. 24
Now the cost of one dozen pencils = cost of 1 pencil multiplied by 12.
So the cost of one dozen pencils =1(12) = Rs. 12
Let one dozen notebook prices be Rs. x.
Therefore, x = 36 Rs.
Let one dozen pen prices be Rs. y.
Therefore, y = 24 Rs.
Let one dozen pencil prices be Rs. z.
Therefore, z = 12 Rs.
So the total cost of items A purchased is
A = 10x + 4y + 5z
B = 8x + 6x + 7z
C = 10x + 5y + 11z
Now convert these into matrix format we have,
$ \Rightarrow \left[ {\begin{array}{*{20}{c}}
  A \\
  B \\
  C
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
  {10}&4&5 \\
  8&6&7 \\
  {10}&5&{11}
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
  x \\
  y \\
  z
\end{array}} \right]$
Now substitute the values of x, y and z we have,
$ \Rightarrow \left[ {\begin{array}{*{20}{c}}
  A \\
  B \\
  C
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
  {10}&4&5 \\
  8&6&7 \\
  {10}&5&{11}
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
  x \\
  y \\
  z
\end{array}} \right]$
$ \Rightarrow \left[ {\begin{array}{*{20}{c}}
  A \\
  B \\
  C
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
  {10}&4&5 \\
  8&6&7 \\
  {10}&5&{11}
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
  {36} \\
  {24} \\
  {12}
\end{array}} \right]$
Now apply matrix multiplication (i.e. first row with first column, second row with second column and so on) so we have,
$ \Rightarrow \left[ {\begin{array}{*{20}{c}}
  A \\
  B \\
  C
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
  {10 \times 36 + 4 \times 24 + 5 \times 12} \\
  {8 \times 36 + 6 \times 24 + 7 \times 12} \\
  {10 \times 36 + 5 \times 24 + 11 \times 12}
\end{array}} \right]$
Now simplify this we have,
$ \Rightarrow \left[ {\begin{array}{*{20}{c}}
  A \\
  B \\
  C
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
  {360 + 96 + 60} \\
  {288 + 144 + 84} \\
  {360 + 120 + 132}
\end{array}} \right]$
Now add these values we have,
$ \Rightarrow \left[ {\begin{array}{*{20}{c}}
  A \\
  B \\
  C
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
  {516} \\
  {516} \\
  {612}
\end{array}} \right]$
So each individual will is
A costs Rs. 516
B costs Rs. 516
C costa Rs. 612.
So this is the required answer.

Note – Whenever we face such types of question the key concept we have to remember is that in matrix multiplication (first row multiply with first column, second row multiply with second column and so on) so first find out the price of every given items per dozen then write the equations of every individual will as above then convert these individual will into matrix format then apply matrix multiplication and simplify as above we will get the required each individual will.