MMT-002 Solved Assignment 2023

IGNOU MMT-002 Solved Assignment 2023 | M.Sc. MACS

Solved By – Narendra Kr. Sharma – M.Sc (Mathematics Honors) – Delhi University

351.00

Share with your Friends

Details For MMT-002 Solved Assignment

IGNOU MMT-002 Assignment Question Paper 2023


Course Code: MMT-002

Assignment Code: MMT-002/TMA/2023

Maximum Marks: 100

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

i) If \(V\) is a finite dimensional vector space and \(T: V \rightarrow V\) 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 \(3 \times 3\) matrix with minimal polynomial \((x-1)^{2}(x-2)\).

iii) If \(\lambda\) is the eigenvalue of a matrix \(A\) with characteristic polynomial \(f(x),(x-\lambda)^{k} \mid f(x)\) and \((x-\lambda)^{k+1} \nmid f(x)\), then the geometric multiplicity of \(\lambda\) is at most \(k\).

iv) If \(\rho(A)=1\), then \(A^{k} \rightarrow \infty\) as \(k \rightarrow \infty\).

v) If \(N\) is nilpotent, \(e^{N}\) 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 \(n \times n\) matrix need not be unique.

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

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

2) a) Let \(T: \mathbf{C}^{2} \rightarrow \mathbf{C}^{2}: T\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{c}x+2 y-i z \\ 2 y+i z \\ i x+z-2 z\end{array}\right]\). Find \([T]_{B},[T]_{B^{\prime}}\) and \(P\) where

\[
B=\left\{\left[\begin{array}{l}
0 \\
i \\
0
\end{array}\right],\left[\begin{array}{c}
i \\
1 \\
-1
\end{array}\right],\left[\begin{array}{l}
0 \\
0 \\
2
\end{array}\right]\right\}, B^{\prime}=\left\{\left[\begin{array}{c}
1 \\
-i \\
1
\end{array}\right],\left[\begin{array}{l}
0 \\
0 \\
1
\end{array}\right],\left[\begin{array}{l}
1 \\
i \\
0
\end{array}\right]\right\},[T]_{B^{\prime}}=P^{-1}[T]_{B} P
\]

b) If \(C\) and \(D\) are \(n \times n\) matrices such that \(C D=-D C\) 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

\(B=\left[\begin{array}{cccc}-1 & 0 & -2 & -4 \\ 2 & 1 & 2 & 4 \\ -4 & 2 & -1 & -4 \\ 2 & -1 & 1 & 3\end{array}\right]\)

Also, find a matrix \(P\) such that \(J=P^{-1} B P\).

4) a) 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 \(\mathrm{M}\) and 10\% of M’s population moves to T. What is the long term effect of on the population of \(\mathrm{M}\) and T? Are they likely to stabilise?

b) Solve the following system of differential equations:

\[
\frac{d y(t)}{d t}=A y(t) \text { with } y(0)=\left[\begin{array}{l}
1 \\
1 \\
1
\end{array}\right], \text { where } A=\left[\begin{array}{ccc}
2 & -5 & -11 \\
0 & -2 & -9 \\
0 & 1 & 4
\end{array}\right]
\]

5) a) Let

\[
A=\left[\begin{array}{ccc}
2 & 2 & 1 \\
-1 & -1 & 2 \\
0 & 0 & -2
\end{array}\right]
\]

Find a unitary matrix \(U\) such that \(U^{*} A U\) 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:

\[
A=\left[\begin{array}{lll}
1 & 1 & 0 \\
1 & 2 & 1 \\
0 & 1 & 1
\end{array}\right], B=\left[\begin{array}{ccc}
2 & 0 & 1 \\
0 & 2 & -1 \\
1 & -1 & 3
\end{array}\right]
\]

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

b) Find the \(\mathrm{QR}\) decomposition of the matrix

\[
\left[\begin{array}{ccc}
2 & -2 & 1 \\
2 & 2 & 1 \\
0 & 1 & 1 \\
1 & 0 & 1
\end{array}\right]
\]

7) Find the SVD of the following matrices:
i) \(\left[\begin{array}{ccc}-1 & 1 & 1 \\ 1 & 1 & 0\end{array}\right]\)
ii) \(\left[\begin{array}{cc}-1 & 1 \\ 1 & 1 \\ 1 & 2\end{array}\right]\)

\(2\:sin\:\theta \:cos\:\phi =sin\:\left(\theta +\phi \right)+sin\:\left(\theta -\phi \right)\)

MMT-002 Sample Solution 2023

 

Frequently Asked Questions (FAQs)

You can access the Complete Solution through our app, which can be downloaded using this link:

App Link 

Simply click “Install” to download and install the app, and then follow the instructions to purchase the required assignment solution. Currently, the app is only available for Android devices. We are working on making the app available for iOS in the future, but it is not currently available for iOS devices.

Yes, It is Complete Solution, a comprehensive solution to the assignments for IGNOU. Valid from January 1, 2023 to December 31, 2023.

Yes, the Complete Solution is aligned with the IGNOU requirements and has been solved accordingly.

Yes, the Complete Solution is guaranteed to be error-free.The solutions are thoroughly researched and verified by subject matter experts to ensure their accuracy.

As of now, you have access to the Complete Solution for a period of 6 months after the date of purchase, which is sufficient to complete the assignment. However, we can extend the access period upon request. You can access the solution anytime through our app.

The app provides complete solutions for all assignment questions. If you still need help, you can contact the support team for assistance at Whatsapp +91-9958288900

No, access to the educational materials is limited to one device only, where you have first logged in. Logging in on multiple devices is not allowed and may result in the revocation of access to the educational materials.

Payments can be made through various secure online payment methods available in the app.Your payment information is protected with industry-standard security measures to ensure its confidentiality and safety. You will receive a receipt for your payment through email or within the app, depending on your preference.

The instructions for formatting your assignments are detailed in the Assignment Booklet, which includes details on paper size, margins, precision, and submission requirements. It is important to strictly follow these instructions to facilitate evaluation and avoid delays.

\(2\:sin\:\theta \:cos\:\phi =sin\:\left(\theta +\phi \right)+sin\:\left(\theta -\phi \right)\)

Terms and Conditions

  • The educational materials provided in the app are the sole property of the app owner and are protected by copyright laws.
  • Reproduction, distribution, or sale of the educational materials without prior written consent from the app owner is strictly prohibited and may result in legal consequences.
  • Any attempt to modify, alter, or use the educational materials for commercial purposes is strictly prohibited.
  • The app owner reserves the right to revoke access to the educational materials at any time without notice for any violation of these terms and conditions.
  • The app owner is not responsible for any damages or losses resulting from the use of the educational materials.
  • The app owner reserves the right to modify these terms and conditions at any time without notice.
  • By accessing and using the app, you agree to abide by these terms and conditions.
  • Access to the educational materials is limited to one device only. Logging in to the app on multiple devices is not allowed and may result in the revocation of access to the educational materials.

Our educational materials are solely available on our website and application only. Users and students can report the dealing or selling of the copied version of our educational materials by any third party at our email address (abstract4math@gmail.com) or mobile no. (+91-9958288900).

In return, such users/students can expect free our educational materials/assignments and other benefits as a bonafide gesture which will be completely dependent upon our discretion.

Scroll to Top
Scroll to Top