You are offered two different sales jobs The first company o

You are offered two different sales jobs. The first company offers a straight commission of 3% of the sales. The second company offers a salary of $240 per week plus 1% of the sales. How much would you have to sell in a week in order for the straight commission offer to be at least as good?

a) Express this problem into a system of two linear equations

b) Determine the rank of the coefficient matrix, the rank of the augmented matrix, and the number of solutions in the problem.

c) Apply Cramer’s rule to determine the solution of the linear system.

d) How much do you have to sell to match both offers? What would be the total income per week? Which offer would you take?

Solution

a) Let X1 is sales for first and second company,

let first company commission is S1and second company salary is S2

hence first company\'s commission would be S1= 0.03X1 ......(1)

second company\'s salary would be S2 = 240+0.01X1 ........(2)

if we match both offer, for 1 week 1st company commission should be equal to 2nd company salary..

S1 = S2 or S1=S2=S

so, We can express equation 1 and 2 in following form.

0.03X - S = 0

-0.01X + S = 240

b.1) coefficient matrix of the above linear equation

Eliminate elements in the 1st column under the 1st element

the number of linearly independent rows are 2

hence the rank of coefficient matrix is 2

b.2) Rank of augmented matrix

add row 2 and row 1, updated matrix as follow

Multiply row 1 by 1/4 and subtract it in row 2, updated matrix

Multiply row 1 by 100/4, updated matrix would be as

Now we have augmented matrix,

Since there is 2 non-zero rows, then Rank(A) = 2.

b.3) Number of solution : The current problem is having unique solution.

c) Solution by Cramer\'s Rule

Write down the main matrix and find its determinant

D = 0.0199

Replace the 1st column of the main matrix with the solution vector and find its determinant

D1 = 240

Replace the 2nd column of the main matrix with the solution vector and find its determinant

D2 = 7.199

x1 = D1 / D = 240 / 0.0199 = 12000.00

x2 = D2 / D = 7.199 / 0.0199 = 360.00

d) Total to sell is 12000,

total income 360 Dollor per week