Linear Algebra Answers

Check whether the following system of equations has a solution.
4x+2y+8z+6w=4
2x+2y+2z+2w=0
x+3z+2w==3

Find the vector equation of the plane determined by point ( 1,1,-1) (1, 1,1) and (0,1,1). Also find point of intersection of line r = (1+3t)I + (2-t) j + (1+t) and the plane.

Let P^(e)= {p(x)€R[x] | p(x)=p(-x) }
P^(o)={p(x)€R[x] | p(x)=-p(-x) }
Check that P^(e) and P^(o) are subspaces of R[x].

Which of the following are subspaces of R^3? Justify your answer.
1) S={(x,y,z)€R^3 | x+y=z }
2) S={(x,y,z)€R^3 | 2x=3yz }

Solve the system of equations 3x+2y+4z=7
2x+y+z=1
x+3y+5z=2
with partial pivoting.Store the multipliers and also write the pivoting vectors.

Check whether the set of vectors v1=(1,1,0,1), v2=(1,0,2,1), v3=(-1,1,-3,-2) €R^4 are linearly independent. If they are dependent, find a1,a2 and a3 €R ,not all zero, such that a1v1+a2v2+a3v3=0.

A company is producing two products, product A and product B. It takes 2 hours too produce one unit of product A and 1 hour to produce 1 unit of product B. To produce one unit of product B it costs 10 rupees. The total budget available is 100 rupees. It is required that the company produce at least 10 units of product A and product B taken together. However, the company can not produce more than 5 units of product B. It is required to find how many units of A and B should be produced so that the total production time is minimized. Formulate the above problem as a linear programming problem and solve it by the graphical method.

An investor wants to invest $30,000 in corporate bonds that are rated AAA, A, and B.

The lower rated ones pay higher interest, but pose a higher risk as well.

The average yield is 5% on AAA, 6% on A bonds, and 10% on B bonds.

Being conservative, the investor wants to have twice as much in AAA bonds as in B bonds.

How much should she invest in each type of bond to have an interest income of $2000?

Find the inverse, if possible, for the following matrices.

(a) (8 -5)
11 23

(b) (5 -7 6)
-11 6 2
2 4 -7

Q2. If A= (1 2 6),
4 11 7
9 13 3

(a) Find the minors of 1,2 and 6.
(b) Find the cofactors of 1,2 and 6.
(c) Evaluate |A|.
(d) A^-1