# Cayley-Hamilton

• 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 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 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)$is a Field and Find the Inverse Elements Let$\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$A^3=O$. 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 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 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 $B=\{1, x, […] • The Product of a Subgroup and a Normal Subgroup is a Subgroup Let$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 […]