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Mathematics Question on Matrices

A manufacturer produces three products x,y,zx,y,z which he sells in two markets. Annual sales are indicated below:MarketProducts
I10000
II6000
(a)If unit sale prices of x,yx,y and zz 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.
Answer

(a) The unit sale prices of x,yx, y, and zz are respectively given as Rs 2.50, Rs 1.50, and Rs 1.00. Consequently, the total revenue in market I can be represented in the form of a matrix as:
[10000200018000][2.50 1.50 1.00]\begin{bmatrix}10000& 2000& 18000\end{bmatrix}\begin{bmatrix}2.50\\\ 1.50\\\ 1.00\end{bmatrix}
=10000×2.50+2000×1.50+18000×1.00]=10000\times2.50+2000\times1.50+18000\times1.00]
=25000+3000+18000=25000+3000+18000
=46000=46000
The total revenue in market II can be represented in the form of a matrix as:
[6000200008000][2.50 1.50 1.00]\begin{bmatrix}6000& 20000& 8000\end{bmatrix}\begin{bmatrix}2.50\\\ 1.50\\\ 1.00\end{bmatrix}
=6000×2.50+20000×1.50+8000×1.00=6000\times2.50+20000\times1.50+8000\times1.00
=15000+30000+8000=15000+30000+8000
=53000=53000
Therefore, the total revenue in market I is Rs 46000 and the same in market II is Rs 53000.

(b) The unit cost prices of x, y, and z are respectively given as Rs 2.00, Rs 1.00, and 50 paise.
Consequently, the total cost prices of all the products in market I can be represented in the form of a matrix as:
[10000200018000][2.00 1.00 0.50]\begin{bmatrix}10000& 2000& 18000\end{bmatrix}\begin{bmatrix}2.00\\\ 1.00\\\ 0.50\end{bmatrix}
=10000×2.00+2000×1.00+18000×0.50=10000\times2.00+2000\times1.00+18000\times0.50
=20000+2000+9000=20000+2000+9000
=31000=31000
Since the total revenue in market I is Rs 46000, the gross profit in this market is (Rs 46000−Rs 31000) Rs 15000.
The total cost prices of all the products in market II can be represented in the form of a matrix as:
[6000200008000][2.00 1.00 0.50]\begin{bmatrix}6000& 20000& 8000\end{bmatrix}\begin{bmatrix}2.00\\\ 1.00\\\ 0.50\end{bmatrix}
=6000×2.00+20000×1.00+8000×0.50=6000\times2.00+20000\times1.00+8000\times0.50
=12000+20000+4000=12000+20000+4000
=Rs36000=Rs 36000
Since the total revenue in market II isRs 53000, the gross profit in this market is (Rs53000 − Rs 36000) Rs 17000.