Question

A sales person Ravi has the following record of sales for the month of January, February and March 2009 for three products A, B and C. He has been paid a commission at fixed rate per unit but at varying rates for products A, B and C.

Months	Sales in units			Commission
	A	B	C
January	9	10	2	800
February	15	5	4	900
March	6	10	3	850

Find the rate of commission payable on A, B and C per unit sold using matrix inversion method.

Answer 1

Let x, y and z be the rate of commission for the three products A, B and C respectively.

9x + 10y + 2z = 800

15x + 5y + 4z = 900

6x + 10y + 3z = 850

The given system can be written as

`[(9,10,2),(15,5,4),(6,10,3)] [(x),(y),(z)] = [(800),(900),(850)]`

AX = B

Where A = `[(9,10,2),(15,5,4),(6,10,3)]`, X = `[(x),(y),(z)]` and B = `[(800),(900),(850)]`

Now, |A| = `[(9,10,2),(15,5,4),(6,10,3)]`

`= 9|(5,4),(10,3)| - 10|(15,4),(6,3)| + 2|(15,5),(6,10)|`

= 9[15 – 40] – 10(45 – 24) + 2(150 – 30)

= 9[-25] – 10[21] + 2[120]

= - 225 – 210 + 240

= - 195

`["A"_"ij"] = [(-25,-21,120),(-|(10,2),(10,3)|,|(9,2),(6,3)|,-|(9,10),(6,10)|),(|(10,2),(5,4)|,-|(9,2),(15,4)|,|(9,10),(15,5)|)]`

`= [(-25,-21,120),(-(30-20),(27-12),-(90-60)),((40 - 10),-(36-30),(45-150))]`

`= [(-25,-21,120),(-10,15,-30),(30,-6,-105)]`

adj A = [A_ij]^T = `[(-25,-10,30),(-21,15,-6),(120,-30,-105)]`

`"A"^-1 = 1/|"A"|` (adj A) = `1/(-195) [(-25,-10,30),(-21,15,-6),(120,-30,-105)]`

X = A^-1B

`[(x),(y),(z)] = (-1)/195 [(-25,-10,30),(-21,15,-6),(120,-30,-105)][(800),(900),(850)]`

= `(-1)/195 [(-20000 - 9000 + 25500),(-16800 + 13500 - 5100),(96000 - 27000 - 89250)]`

`=> (-1)/195[(-3500),(-8400),(-20250)]`

`[(x),(y),(z)] = [(17.948),(43.0769),(103.846)]`

`[(x),(y),(z)] = [(17.95),(43.08),(103.85)]`

∴ x = 17.95, y = 43.08, z = 103.85

The rate of commission of A, B and C are 17.95, 43.08 and 103.85 respectively.