IGNOU MMT 2 SOLVED ASSIGNMENT

MMT 2: Linear Algebra

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Title Name	IGNOU MMT 2 SOLVED ASSIGNMENT
Type	Soft Copy (E-Assignment) .pdf
University	IGNOU
Degree	MASTER DEGREE PROGRAMMES
Course Code	MSCMACS
Course Name	M.Sc. Mathematics with Applications in Computer Science
Subject Code	MMT 2
Subject Name	Linear Algebra
Year	2026
Session	-
Language	English Medium
Assignment Code	MMT 2/Assignment-1/2026
Product Description	Assignment of MSCMACS (M.Sc. Mathematics with Applications in Computer Science) 2026. Latest MMT 02 2026 Solved Assignment Solutions
Last Date of IGNOU Assignment Submission	Last Date of Submission of IGNOU BEGC-131 (BAG) 2025-26 Assignment is for January 2026 Session: 30th September, 2026 (for December 2025 Term End Exam). Semester Wise January 2025 Session: 30th March, 2026 (for June 2026 Term End Exam). July 2025 Session: 30th September, 2025 (for December 2025 Term End Exam).
Format	Ready-to-Print PDF (.soft copy)

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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 $T:V ightarrow V$ 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 - $lambda$ )^k | f(x) and $(x-lambda)^{k+1}+f(x)$ then the geometric multiplicity of a is at most k.

iv) If $ho(A)$ = 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 $T:mathbb{C}^2 omathbb{C}^2:Tegin{bmatrix}x\y\zend{bmatrix}=egin{bmatrix}x+2y-iz\2y+iz\ix+z-2zend{bmatrix}$ Find [T]_B [T]_B and P where

$B=left{egin{bmatrix}0\i\0end{bmatrix},egin{bmatrix}i\1\-1end{bmatrix},egin{bmatrix}0\0\2end{bmatrix} ight},quad B'=left{egin{bmatrix}1\-i\1end{bmatrix},egin{bmatrix}0\0\1end{bmatrix},egin{bmatrix}1\i\0end{bmatrix} ight},quad[T]_{B'}=P^{-1}[T]_B P.$

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

$B=egin{bmatrix}-1&0&-2&-4\2&1&2&4\-4&2&-1&-4\2&-1&1&3end{bmatrix}$

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:

$frac{dy(t)}{dt}=Ay(t) ext{with}y(0)=egin{bmatrix}1\1\1end{bmatrix}, ext{where}A=egin{bmatrix}2&-5&-11\0&-2&-9\0&1&4end{bmatrix}$

5) a) Let

$A=egin{bmatrix}2&2&1\-1&-1&2\0&0&-2end{bmatrix}$

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:

$A=egin{bmatrix}1&1&0\1&2&1\0&1&1end{bmatrix},quad B=egin{bmatrix}2&0&1\0&2&-1\1&-1&3end{bmatrix}$

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

b) Find the QR decomposition of the matrix

$egin{bmatrix}2&-2&1\2&2&1\0&1&1\1&0&1end{bmatrix}$

7) Find the SVD of the following matrices:

(i) $egin{bmatrix}-1&1&1\1&1&0end{bmatrix}$ (ii) $egin{bmatrix}-1&1\1&1\1&2end{bmatrix}$

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Course Name	M.Sc. Mathematics with Applications in Computer Science
Course Code	MSCMACS
Programm	Courses
Language	English

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