Solve the linear system by row-reducing the corresponding

Question

Q-1:

Answer each of the following as True or False (justify your answer):

a) If X₁ and X₂ are solutions of the equation AX = B (B ≠ 0), then X₁ + X₂ is also a solution.

b) If A andB are 2 x 2 matrices, then the sum of the terms on the main diagonal of AB - BA is zero.

c) SupposeA is a 2 x 2 matrix. If A is invertible, then At is also invertible.

c) Suppose that S = {v₁, v₂, v₃} is a linearly independent set in Rn; then T = {v₁, v₂, v₁+ v₂+ v₃} is also linearly independent.

e) For a ∈ R, the vectors are linearly independent.

Q-2:

Let

 

	-1	1	0	-1
	-1	1	-1	0
A =	-1	0	0	0
	-2	1	-1	1

a) Find A^-1.

b) Solve the linear system AX = B = , where

Q-3: Consider the linear system Ax = b where

a) Solve the linear system;

b) Give a particular solution S_p;

c) Solve the homogeneous equation Ax = 0.

Q4: Let

a) Find a matrix B in reduced-row echelon form that is row equivalent to A;
b) Find det(A);
c) Calculate det(1/3A^-1A^t);

Q5: Consider the linear system

x - 6y - 4z = -5

2x - 10y - 9z = -4

-x + 6y + 5z = 3

a) Solve the linear system by row-reducing the corresponding augmented matrix;

b) Find the values of h for which the following set of vectors is linearly independent:

Q6: Given the vectors 

a) Show that the set T = {v₁, v₂, v₃ ,v₄} is linearly dependent;

b) Determine whether the set S = {v₁, v₂ ,v₃} is linearly dependent;

c) Find the scalars such that v₄ can be written as v₄ = c₁v₁ + c₂v₂ + c₃v₃;

d) Find all vectors w = that can be written as a linear combination of the set U = {v₂, v₃ ,v₄}.