Cayley-Hamilton

Cayley-Hamilton

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Cayley-Hamilton Theorem Problems and Solutions


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  • Is the Product of a Nilpotent Matrix and an Invertible Matrix Nilpotent?Is the Product of a Nilpotent Matrix and an Invertible Matrix Nilpotent? A square matrix $A$ is called nilpotent if there exists a positive integer $k$ such that $A^k=O$, where $O$ is the zero matrix. (a) If $A$ is a nilpotent $n \times n$ matrix and $B$ is an $n\times n$ matrix such that $AB=BA$. Show that the product $AB$ is nilpotent. (b) Let $P$ […]
  • Probability that Alice Tossed a Coin Three Times If Alice and Bob Tossed Totally 7 TimesProbability that Alice Tossed a Coin Three Times If Alice and Bob Tossed Totally 7 Times Alice tossed a fair coin until a head occurred. Then Bob tossed the coin until a head occurred. Suppose that the total number of tosses for Alice and Bob was $7$. Assuming that each toss is independent of each other, what is the probability that Alice tossed the coin exactly three […]
  • A Line is a Subspace if and only if its $y$-Intercept is ZeroA Line is a Subspace if and only if its $y$-Intercept is Zero Let $\R^2$ be the $x$-$y$-plane. Then $\R^2$ is a vector space. A line $\ell \subset \mathbb{R}^2$ with slope $m$ and $y$-intercept $b$ is defined by \[ \ell = \{ (x, y) \in \mathbb{R}^2 \mid y = mx + b \} .\] Prove that $\ell$ is a subspace of $\mathbb{R}^2$ if and only if $b = […]
  • Prove that $\F_3[x]/(x^2+1)$ is a Field and Find the Inverse ElementsProve that $\F_3[x]/(x^2+1)$ is a Field and Find the Inverse Elements Let $\F_3=\Zmod{3}$ be the finite field of order $3$. Consider the ring $\F_3[x]$ of polynomial over $\F_3$ and its ideal $I=(x^2+1)$ generated by $x^2+1\in \F_3[x]$. (a) Prove that the quotient ring $\F_3[x]/(x^2+1)$ is a field. How many elements does the field have? (b) […]
  • Example of a Nilpotent Matrix $A$ such that $A^2\neq O$ but $A^3=O$.Example of a Nilpotent Matrix $A$ such that $A^2\neq O$ but $A^3=O$. Find a nonzero $3\times 3$ matrix $A$ such that $A^2\neq O$ and $A^3=O$, where $O$ is the $3\times 3$ zero matrix. (Such a matrix is an example of a nilpotent matrix. See the comment after the solution.)   Solution. For example, let $A$ be the following $3\times […]
  • Every Group of Order 12 Has a Normal Subgroup of Order 3 or 4Every Group of Order 12 Has a Normal Subgroup of Order 3 or 4 Let $G$ be a group of order $12$. Prove that $G$ has a normal subgroup of order $3$ or $4$.   Hint. Use Sylow's theorem. (See Sylow’s Theorem (Summary) for a review of Sylow's theorem.) Recall that if there is a unique Sylow $p$-subgroup in a group $GH$, then it is […]
  • Vector Space of Polynomials and a Basis of  Its SubspaceVector Space of Polynomials and a Basis of Its Subspace Let $P_2$ be the vector space of all polynomials of degree two or less. Consider the subset in $P_2$ \[Q=\{ p_1(x), p_2(x), p_3(x), p_4(x)\},\] where \begin{align*} &p_1(x)=1, &p_2(x)=x^2+x+1, \\ &p_3(x)=2x^2, &p_4(x)=x^2-x+1. \end{align*} (a) Use the basis $B=\{1, x, […]
  • The Product of a Subgroup and a Normal Subgroup is a SubgroupThe Product of a Subgroup and a Normal Subgroup is a Subgroup Let $G$ be a group. Let $H$ be a subgroup of $G$ and let $N$ be a normal subgroup of $G$. The product of $H$ and $N$ is defined to be the subset \[H\cdot N=\{hn\in G\mid h \in H, n\in N\}.\] Prove that the product $H\cdot N$ is a subgroup of […]

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