# pi2018

by Yu ·

Add to solve later

Add to solve later

Add to solve later

### More from my site

- Number Theoretical Problem Proved by Group Theory. $a^{2^n}+b^{2^n}\equiv 0 \pmod{p}$ Implies $2^{n+1}|p-1$. Let $a, b$ be relatively prime integers and let $p$ be a prime number. Suppose that we have \[a^{2^n}+b^{2^n}\equiv 0 \pmod{p}\] for some positive integer $n$. Then prove that $2^{n+1}$ divides $p-1$. Proof. Since $a$ and $b$ are relatively prime, at least one […]
- Compute Power of Matrix If Eigenvalues and Eigenvectors Are Given Let $A$ be a $3\times 3$ matrix. Suppose that $A$ has eigenvalues $2$ and $-1$, and suppose that $\mathbf{u}$ and $\mathbf{v}$ are eigenvectors corresponding to $2$ and $-1$, respectively, where \[\mathbf{u}=\begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix} \text{ […]
- Is the Determinant of a Matrix Additive? Let $A$ and $B$ be $n\times n$ matrices, where $n$ is an integer greater than $1$. Is it true that \[\det(A+B)=\det(A)+\det(B)?\] If so, then give a proof. If not, then give a counterexample. Solution. We claim that the statement is false. As a counterexample, […]
- Is the Set of Nilpotent Element an Ideal? Is it true that a set of nilpotent elements in a ring $R$ is an ideal of $R$? If so, prove it. Otherwise give a counterexample. Proof. We give a counterexample. Let $R$ be the noncommutative ring of $2\times 2$ matrices with real […]
- True or False: $(A-B)(A+B)=A^2-B^2$ for Matrices $A$ and $B$ Let $A$ and $B$ be $2\times 2$ matrices. Prove or find a counterexample for the statement that $(A-B)(A+B)=A^2-B^2$. Hint. In general, matrix multiplication is not commutative: $AB$ and $BA$ might be different. Solution. Let us calculate $(A-B)(A+B)$ as […]
- A Linear Transformation is Injective (One-To-One) if and only if the Nullity is Zero Let $U$ and $V$ be vector spaces over a scalar field $\F$. Let $T: U \to V$ be a linear transformation. Prove that $T$ is injective (one-to-one) if and only if the nullity of $T$ is zero. Definition (Injective, One-to-One Linear Transformation). A linear […]
- The Order of $ab$ and $ba$ in a Group are the Same Let $G$ be a finite group. Let $a, b$ be elements of $G$. Prove that the order of $ab$ is equal to the order of $ba$. (Of course do not assume that $G$ is an abelian group.) Proof. Let $n$ and $m$ be the order of $ab$ and $ba$, respectively. That is, \[(ab)^n=e, […]
- Rank and Nullity of Linear Transformation From $\R^3$ to $\R^2$ Let $T:\R^3 \to \R^2$ be a linear transformation such that \[ T(\mathbf{e}_1)=\begin{bmatrix} 1 \\ 0 \end{bmatrix}, T(\mathbf{e}_2)=\begin{bmatrix} 0 \\ 1 \end{bmatrix}, T(\mathbf{e}_3)=\begin{bmatrix} 1 \\ 0 \end{bmatrix},\] where $\mathbf{e}_1, […]