Mathematics II 2067

Attempt all questions:

Group A (10x2=20)

1. Illustrate by an example that a system of linear equations has either no solution or exactly

one solution.

2. Define singular and nonsingular matrices.

3. Using the Invertible matrix Theorem or otherwise, show that  is invertible.

4. What is numerical drawback of the direct calculation of the determinants?

5. Verify with an example that det (AB ) = det ( A) det ( B) for any n x n matrices A and B.

6. Find a matrix A such that col(A).

7. Define subspace of a vector with an example.

8. Are the vectors;  eigen vectors of  

9. Find the distance between vectors u ( ) and v ( ). Define the distance between

two vectors in  Rⁿ. 

10. Let w = span {x₁, x₂}, where  Then construct orthogonal basis for w.

Group B (5x4=20)

11. If a set s = {v₁, v_{2,  ... ... ... ,}v_p} in Rⁿ contains the zero vector, then prove that the set is linearly dependent. Determine if the set  is linearly dependent.

12. Given the Leontief input-output model x = Cx + d, where the symbols have their usual

meanings, consider any economy whose consumption matrix is given by Suppose the final demand is 50 units for manufacturing 30 units for agriculture, 20 units for services. Find the production level x that will satisfy this demand.

13. Define rank of a matrix and state Rank Theorem. If A is a 7 x 9 matrix with a

two-dimensional null space, find the rank of A.

14. Determine the eigen values and eigen vectors of   in complex numbers.

OR

Let and basis B = {b₁, b₂}.Find the B-matrix for the transformation  with P= [b₁, b₂].

15. Let u and v be nonzero vectors in R² and the angle between them be θ then prove that u.v = ‖u‖ ‖v‖ cos θ,

where the symbols have their usual meanings.

Group C (8 x 5 = 40)

16. Determine if the following homogeneous system has a nontrivial solution. Then describe the

solution set. 3x₁ + 5x₂ - 4x₃ = 0, - 3x₁ -2x + 4x₃= 0, 6x₁ + x₂ - 8x₃ = 0.

17. An n x n matrix A is invertible if and only if A is row equivalent to I_n , and in this case, any sequence of elementary row operations that reduces A to I_n also transform I_{n x m} into A^-1. Use this statement to find the inverse of the matrix if exist.

18. What do you mean by basis change? Consider two bases B = {b₁, b₂} and c = {c₁, c₂} for a vector space V, such that  and Suppose  i.e., x = 3b₁ + b₂. Find [x]_c.

OR

Define basis of a subspace of a vector space.Let  where   and let  Show that span {v₁, v₂, v₃} = span {v₁, v₂}  and find a basis for the subspace H.

19. Diagonalize the matrix if possible.

20. What do you mean by least-squares lines? Find the equation  of the least- squares line that fits the data points (2, 1), (5, 2), (7, 3), (8, 3).

OR

Find the least-squares solution of Ax = b for A