Mathematics II 2066

Attempt all questions:

Group A (10 x 2 = 20)

1. When is system of linear equation consistent or inconsistent?

2. Write numerical importance of partitioning matrices.

3. How do you distinguish singular and non-singular matrices?

4. If A and B are n x n matrices, then verify with an example that det(AB) = det(A)det(B).

5. Calculate the area of the parallelogram determined by the columns of 

7. Determine if {v₁, v₂, v₃} is a basis for  , where 

6. Determine if  is Nul(A), where, 

8. Find the characteristic polynomial for the eigen values of the matrix 

9. Let  Find a unit vector  in the same direction as 

10. Let {u₁,... ... ... u_p}be an orthogonal basis for a subspace W of Rⁿ. Then prove that for each ,the weights in y = c₁u₁ + ... ... ... + c_pu_p are given by

Group B (5 x 4 = 20)

11. Prove that any set{v₁,... ... ... ... , v₂} in Rⁿ is linearly dependent if p > n.

12. Consider the Leontief input – output model equation x = cx + d, where the consumption matrix is

 Suppose the final demand is 50 units of manufacturing, 30 units of agriculture, 20 units for services. Find the production level x that will satisfy the demand.

13. What do you mean by basis of a vector space? Find the basis for the row space of 

OR

State and prove the unique representation theorem for coordinate systems.

14. What do you mean by eigen values, eigen vectors and characteristic polynomial of a matrix? Explain with suitable examples.

15. Define the Gram-Schmidt process.  Let W=span{x₁, x₂}, where  Then construct an orthogonal basis {v₁, v₂} for w.

Group C (5 x 8 = 40)

16. Given the matrix discuss the for word phase and backward phase of the row reduction algorithm.

17. Find the inverse of  if it exists, by using elementary row reduce the augmented matrix.

18. What do you mean by change of basis in Rⁿ ? Let and consider the bases for R² given by B={b₁, b₂} and C={c₁, c₂}. Find the change of coordinates matrix from B to C.

19. Diagonalize the matrix  if possible 

OR

Find the eigen value of   and find a basis for each eigen space.

20. Find a least-square solution for Ax = b with  What do you mean by least squares problems? 

OR

Define a least-squares solution of Ax = b, prove that the set of least squares solutions of Ax = b coincides with the non-empty set of solutions of the normal equations A^TAx = A^Tb.