Ques 1.

Which of the following statements are true and which are false? Give reasons for your answer

i) If  is a finite dimensional vector space and  is a diagonalisable linear operator, then there is a basis, unique up to order of the elements, with respect to which the matrix of 
is diagonal.
ii) Up to similarity, there is a unique  matrix with minimal polynomial .
iii) If  is the eigenvalue of a matrix

Ques 2.

Let .  FInd [T]_B ,  [T]_B' and P where

, ,

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Ques 3.

If C and D are  matrices such that CD = -DC and D^-1 exists, then show that C is similar to -D. Hence show that the eigenvalues of C must come in plus-minus pairs.

Ques 4.

Can A be similar to A + I? Give reasons for your answer

Ques 5.

Find the Jordan canonical form J for

Also, find a matrix P such that J = P^-1BP

Ques 6.

Let M and T be a metro city and a nearby district town, respectively. Our government is trying to develop infrastructure in T so that people shift to T. Each year 15% of T’s population moves to M and 10% of M’s population moves to T. What is the long term effect of on the population of M and T? Are they likely to stabilise?

Ques 7.

Solve the following system of differential equations:

 with , Where 

Ques 8.

Let

Find a unitary matrix such that  is upper triangular.

Ques 9.

Use least squares method to find a quadratic polynomial that fits the following data:  (-2, 15.7), (-1, 6.7), (0, 2.7), (1, 3.7), (2, 9.7).

Ques 10.

Check which of the following matrices is positive definite and which is positive semi-definite:

Also, find the square root of the positive definite matrix.

Ques 11.

Find the QR decomposition of the matrix

Ques 12.

Find the SVD of the following matrices:

i). 

ii).