Linear Algebra Answers

Consider the following system of equations:

1)

x + 3y + 5z = b1

2x + 4y + 6z = b2.

Show that there are no solutions if b2 = 2b1 and infinitely many solutions otherwise.

2)

x + 4y = b1

2x + 5y = b2

3x + 6y = b3.

Show that there is exactly one solution if b3 = 7b1 − 2b2 and no solutions otherwise.

3. A firm can produce a good either by (i) a labor intensive technique, using 8 units of labor and 1 unit of capital or (ii) a capital intensive technique using 1 unit of labor and 2 units of capital. The firm can arrange up to 200 units of labor and 100 units of capital. Note that the firm produce goods X and Y in process land 1 and 2 respectively. Its objective is to maximize profit by selling the good.

i) Construct the objective function and the constant inequalities.

ii) By drawing the graphs of the linear constraints, find the optimal solutions.

 iii) Find the solution using the simplex methods.

. Work Out

1. Solve by crammer's rule, the following equations.

3X + 3Y - Z = 11

2X - Y + 2Z = 9

4X + 3Y - 27 = 25

2. solve the above equations using graphical method. 

Suppose that a 4 × 4 matrix A has eigenvalues λ₁ = 1, λ₂ = − 5, λ₃ = 6, and λ₄ = − 6. Use the following method to find tr(A).

If A is a square matrix and p(λ) = det (λI − A) is the characteristic polynomial of A, then the coefficient of λ^{n − 1} in p(λ) is the negative of the trace of A.