Linear Algebra Answers

A and B are real non-zero 3 × 3 matrices and satisfy the equation (AB)^T + B^(-1) A = 0. Prove that if B is orthogonal then A is antisymmetric.

A worker requires 8 tanks of crude oil and petrol to get a work done. He gains 5000 

Ghana cedis per tank of crude oil used and 3000 Ghana cedis per petrol used for any 

work done. To finish the work, he further requires 10 tanks by combining 2 tanks of 

crude oil and a tank of petrol. What is the maximum profit he can make?

solve the system of three variable linear equations.

4x - 2y + 3z = 1

x + 3y - 4z = -7

3x + y + 2z = 5

solve the system of three variable linear equations.

x = 3z - 5

2x + 2z = y + 16

7x - 5z = 3y + 19

solve the system of linear equations.

x-y + 2z = 5

2x - 2y + 4z = 10

3x - 3y + 6z = 15

solve the system of three variable linear equations

x + 2y = 1

3x + 2y + 4z = 7

-2x + y - 2z = -1

Determine the polynomial function whose graph passes through the points (0, 1), (2,1/3) and (4,1/5). Also sketch the graph of the polynomial function. 

Let T1 and T2 be linear operators on V such that the range of T2 is contained in the kernel of T1.

Show that T1 ◦ T2 = 0. Also show that rank(T1)+rank(T2) ≤ dim(V )

Let V be the set of all vectors of the form (x1, x2, x3) in R

3

(a) x1 − 3x2 + 2x3 = 0.

(b) 3x1 − 2x2 + x3 = 0 and 4x1 + 5x2 = 0.

Find the dimension and basis for V.

 Let f : A → B be a one-to-one correspondence.

1. Prove that f^-1 is a function.

2. Prove that f^-1 is one-to-one.

3. Prove that f^-1 is onto.

4. Conclude that f^-1 : B → A is a one-to-one correspondence.