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

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

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## 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) $$\left[\begin{array}{cc}-1 & 1 \\ 1 & 1 \\ 1 & 2\end{array}\right]$$

$$a=b\:cos\:C+c\:cos\:B$$

## MMT-002 Sample Solution 2023

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$$cos\:2\theta =1-2\:sin^2\theta$$

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