Reference no: EM131295520
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
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-1 |
1 |
0 |
-1 |
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-1 |
1 |
-1 |
0 |
| A = |
-1 |
0 |
0 |
0 |
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-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 Sp;
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-1At);
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}.
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