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• Determine a Matrix From Its Eigenvalue Let \[A=\begin{bmatrix} a & -1\\ 1& 4 \end{bmatrix}\] be a $2\times 2$ matrix, where $a$ is some real number. Suppose that the matrix $A$ has an eigenvalue $3$. (a) Determine the value of $a$. (b) Does the matrix $A$ have eigenvalues other than […]
• A Matrix Having One Positive Eigenvalue and One Negative Eigenvalue Prove that the matrix \[A=\begin{bmatrix} 1 & 1.00001 & 1 \\ 1.00001 &1 &1.00001 \\ 1 & 1.00001 & 1 \end{bmatrix}\] has one positive eigenvalue and one negative eigenvalue. (University of California, Berkeley Qualifying Exam Problem)   Solution. Let us put […]
• The Center of a p-Group is Not Trivial Let $G$ be a group of order $|G|=p^n$ for some $n \in \N$. (Such a group is called a $p$-group.) Show that the center $Z(G)$ of the group $G$ is not trivial.   Hint. Use the class equation. Proof. If $G=Z(G)$, then the statement is true. So suppose that $G\neq […]
• Condition that a Function Be a Probability Density Function Let $c$ be a positive real number. Suppose that $X$ is a continuous random variable whose probability density function is given by \begin{align*} f(x) = \begin{cases} \frac{1}{x^3} & \text{ if } x \geq c\\ 0 & \text{ if } x < […]
• The Number of Elements Satisfying $g^5=e$ in a Finite Group is Odd Let $G$ be a finite group. Let $S$ be the set of elements $g$ such that $g^5=e$, where $e$ is the identity element in the group $G$. Prove that the number of elements in $S$ is odd.   Proof. Let $g\neq e$ be an element in the group $G$ such that $g^5=e$. As […]
• 10 True or False Problems about Basic Matrix Operations Test your understanding of basic properties of matrix operations. There are 10 True or False Quiz Problems. These 10 problems are very common and essential. So make sure to understand these and don't lose a point if any of these is your exam problems. (These are actual exam […]
• Ring Homomorphisms and Radical Ideals Let $R$ and $R'$ be commutative rings and let $f:R\to R'$ be a ring homomorphism. Let $I$ and $I'$ be ideals of $R$ and $R'$, respectively. (a) Prove that $f(\sqrt{I}\,) \subset \sqrt{f(I)}$. (b) Prove that $\sqrt{f^{-1}(I')}=f^{-1}(\sqrt{I'})$ (c) Suppose that $f$ is […]
• Find All Matrices $B$ that Commutes With a Given Matrix $A$: $AB=BA$ Let \[A=\begin{bmatrix} 1 & 3\\ 2& 4 \end{bmatrix}.\] Then (a) Find all matrices \[B=\begin{bmatrix} x & y\\ z& w \end{bmatrix}\] such that $AB=BA$. (b) Use the results of part (a) to exhibit $2\times 2$ matrices $B$ and $C$ such that \[AB=BA \text{ and } […]