A manufacturer produces three products x, y, z which he sells in two markets. Annual sales are indicated as :

(a) If unit sale prices of x,y and z are Rs 2.50, Rs 1.50 and Rs 1.00, respectively, find the total revenue in each market with the help of matrix algebra.

(b) If the unit costs of the above three commodities are Rs 2.00, Rs 1.00 and 50 paise respectively. Find the gross profit

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.:

Let quantity matrix be $A = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{10000}&{2000}&{18000}\\{6000}&{20000}&{8000}\end{array}} \right]$

(a) Selling Price $B = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{2.50}\\{1.50}\\{1.00}\end{array}} \right]$

Now, Total Selling Price,
$AB = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{10000}&{2000}&{18000}\\{6000}&{20000}&{8000}\end{array}} \right]$ $\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{2.50}\\{1.50}\\{1.00}\end{array}} \right]$

$= \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{10,000 \times 2.50 + 2,000 \times 1.50 + 18,000 \times 1}\\{6,000 \times 2.50 + 20,000 \times 1.50 + 8,000 \times 1}\end{array}} \right]$

$= \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{25,000 + 3,000 + 18,000}\\{15,000 + 30,000 + 8,000}\end{array}} \right]$

Total revenue in market I = Rs. 46,000.

Total revenue in market II = Rs. 53,000.

(b) Now, cost price$= \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{2.00}\\{1.00}\\{0.50}\end{array}} \right]$

Total cost price$= \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{1000}&{2000}&{18000}\\{6000}&{20000}&{8000}\end{array}} \right]\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}2\\1\\{0.5}\end{array}} \right]$

$= \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{10,000 \times 2 + 2,000 \times 1 + 18,000 \times 0.5}\\{6,000 \times 2 + 20,000 \times 1 + 8,000 \times 0.5}\end{array}} \right]$

Total cost price = 31000 + 36000 = Rs. 67,000.

Total selling price = 46000 + 53000 = Rs. 99,000

Profit = S.P.$-$C.P. = 99,000 $-$

67,000 = Rs. 32,000.