IGNOU MMT002 Solved Assignment 2023  M.Sc. MACS
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IGNOU MMT002 Assignment Question Paper 2023
Course Code: MMT002
Assignment Code: MMT002/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 \((x1)^{2}(x2)\).
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 nonnegative.
x) The SVD of any \(2 \times 3\) matrix is unique.
2) a) Let \(T: \mathbf{C}^{2} \rightarrow \mathbf{C}^{2}: T\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{c}x+2 yi z \\ 2 y+i z \\ i x+z2 z\end{array}\right]\). 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 plusminus 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 semidefinite:
\[
A=\left[\begin{array}{lll}
1 & 1 & 0 \\
1 & 2 & 1 \\
0 & 1 & 1
\end{array}\right], B=\left[\begin{array}{ccc}
2 & 0 & 1 \\
0 & 2 & 1 \\
1 & 1 & 3
\end{array}\right]
\]
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) \(\left[\begin{array}{cc}1 & 1 \\ 1 & 1 \\ 1 & 2\end{array}\right]\)
MMT002 Sample Solution 2023
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