IGNOU MMT 2 SOLVED ASSIGNMENT

Included:

MMT 2 2025 - English

Course Code: MMT-002

Assignment Code: MMT-002/TMA/2025

Maximum Marks: 100

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

i) If Vis 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 T is diagonal.

ii) Up to similarity, there is a unique 3 x 3 matrix with minimal polynomial (x - 1)²(x - 2).

iii) If a is the eigenvalue of a matrix A with characteristic polynomial f(x), (x - )^k | f(x) and  then the geometric multiplicity of a is at most k.

iv) If = 1, then A^k → ∞ as k → ∞ .

v) If N is nilpotent, eⁿ is also nilpotent.

vi) The sum of two normal matrices of the order n is normal.

vii) If P and Q are positive definite operators, P + Q is a positive definite operator.

viii) Generalised inverse of a n x n matrix need not be unique.

ix) All the entries of a positive definite matrix are non-negative.

x) The SVD of any 2 x 3 matrix is unique.

2) a) Let Find [T]_B [T]_B and P where

b) If C and D are n x n matrices such that C  = -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.

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

3) Find the Jordan canonical form J for

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

4) a) Let M and Tbe 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?

b) Solve the following system of differential equations:

5) a) Let

Find a unitary matrix U such that U* AU is upper triangular.

b) 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).

6) a) 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.

b) Find the QR decomposition of the matrix

7) Find the SVD of the following matrices:

(i)             (ii)   

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