todai-math

Papers (28)
2025
problems1-3 3 todai-engineering-math 6
2024
problem1 1 problem2 1 problem3 1 todai-engineering-math 6
2023
problem1 1 problem2 1 problem3 1 todai-engineering-math 6
2022
problem1 1 problem2 1 problem3 1 todai-engineering-math__paper1 2 todai-engineering-math__paper2 2 todai-engineering-math__paper3 7
2021
todai-engineering-math__paper1 6 todai-engineering-math__paper2 3 todai-engineering-math__paper3 3
2020
todai-engineering-math 6
2019
todai-engineering-math 6
2018
ist 3 todai-engineering-math 6
2017
ist 3 todai-engineering-math 6
2016
todai-engineering-math 6 translated 3
2015
ist 3
2021 todai-engineering-math__paper2

3 maths questions

QI 3x3 Matrices Matrix Algebraic Properties and Abstract Reasoning View
Answer the following questions concerning the matrix $\boldsymbol{A}$ given by
$$A = \left( \begin{array}{ccc} 0 & 3 & 0 \\ -3 & 0 & 4 \\ 0 & -4 & 0 \end{array} \right)$$
In the following, $\boldsymbol{I}$ is the $3 \times 3$ identity matrix, $\boldsymbol{O}$ is the $3 \times 3$ zero matrix, $n$ is an integer greater than or equal to 0 and $t$ is a real number.
  1. Obtain all eigenvalues of the matrix $\boldsymbol{A}$.
  2. Find coefficients $a, b$ and $c$ of the following equation satisfied by $\boldsymbol{A}$: $$\boldsymbol{A}^3 + a\boldsymbol{A}^2 + b\boldsymbol{A} + c\boldsymbol{I} = \boldsymbol{O}$$
  3. Obtain $\boldsymbol{A}^{2n+1}$.
  4. Since Equation (2) is satisfied, the following equation holds: $$\exp(t\boldsymbol{A}) = p\boldsymbol{A}^2 + q\boldsymbol{A} + r\boldsymbol{I}$$ Express coefficients $p, q$ and $r$ in terms of $t$ without using the imaginary unit.
QII Matrices Eigenvalue and Characteristic Polynomial Analysis View
Consider a discrete-time system where stochastic transitions between the two states (A and B) occur as shown in Figure 2.1. The transition probability in unit time from the state A to B is $\alpha$ and from the state B to A is $\beta$. Note that $0 < \alpha < 1$ and $0 < \beta < 1$. Variables $n$ and $k$ represent discrete time and are integers greater than or equal to 0.
Answer the following questions.
  1. Let $P_{\mathrm{A}}(n)$ be the probability that the state is A at time $n$ and $P_{\mathrm{B}}(n)$ be the probability that the state is B at time $n$. Let $\boldsymbol{P}(n) = \binom{P_{\mathrm{A}}(n)}{P_{\mathrm{B}}(n)}$. Express matrix $\boldsymbol{M}$ using $\alpha$ and $\beta$, assuming $\boldsymbol{P}(n+1) = \boldsymbol{M}\boldsymbol{P}(n)$.
  2. Obtain all eigenvalues and the corresponding eigenvectors of matrix $\boldsymbol{M}$.
  3. As time tends towards infinity, the probability that the state is A and the probability that the state is B converge towards constant values. Obtain each value.
  4. Assume $R_{\mathrm{A}}(n) = P_{\mathrm{A}}(n) - \lim_{k \rightarrow \infty} P_{\mathrm{A}}(k)$. Express $R_{\mathrm{A}}(n+1)$ by using $R_{\mathrm{A}}(n)$.
QIII Matrices Determinant and Rank Computation View
Assume vectors $\boldsymbol{a}_1, \boldsymbol{a}_2, \ldots, \boldsymbol{a}_m$ are linearly independent in a vector space $V$, where $m$ is an integer greater than or equal to 3. Obtain the condition that $m$ must satisfy in order for $\boldsymbol{a}_1 + \boldsymbol{a}_2,\ \boldsymbol{a}_2 + \boldsymbol{a}_3,\ \ldots,\ \boldsymbol{a}_{m-1} + \boldsymbol{a}_m$ and $\boldsymbol{a}_m + \boldsymbol{a}_1$ to be linearly independent.