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• 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 < […]
• Equivalent Conditions to be a Unitary Matrix A complex matrix is called unitary if $\overline{A}^{\trans} A=I$. The inner product $(\mathbf{x}, \mathbf{y})$ of complex vector $\mathbf{x}$, $\mathbf{y}$ is defined by $(\mathbf{x}, \mathbf{y}):=\overline{\mathbf{x}}^{\trans} \mathbf{y}$. The length of a complex vector […]
• Two Matrices are Nonsingular if and only if the Product is Nonsingular An $n\times n$ matrix $A$ is called nonsingular if the only vector $\mathbf{x}\in \R^n$ satisfying the equation $A\mathbf{x}=\mathbf{0}$ is $\mathbf{x}=\mathbf{0}$. Using the definition of a nonsingular matrix, prove the following statements. (a) If $A$ and $B$ are $n\times […]
• What is the Probability that All Coins Land Heads When Four Coins are Tossed If…? Four fair coins are tossed. (1) What is the probability that all coins land heads? (2) What is the probability that all coins land heads if the first coin is heads? (3) What is the probability that all coins land heads if at least one coin lands […]
• Conditions on Coefficients that a Matrix is Nonsingular (a) Let $A=(a_{ij})$ be an $n\times n$ matrix. Suppose that the entries of the matrix $A$ satisfy the following relation. \[|a_{ii}|>|a_{i1}|+\cdots +|a_{i\,i-1}|+|a_{i \, i+1}|+\cdots +|a_{in}|\] for all $1 \leq i \leq n$. Show that the matrix $A$ is nonsingular. (b) Let […]
• Determine whether the Matrix is Nonsingular from the Given Relation Let $A$ and $B$ be $3\times 3$ matrices and let $C=A-2B$. If \[A\begin{bmatrix} 1 \\ 3 \\ 5 \end{bmatrix}=B\begin{bmatrix} 2 \\ 6 \\ 10 \end{bmatrix},\] then is the matrix $C$ nonsingular? If so, prove it. Otherwise, explain why not. […]
• A Prime Ideal in the Ring $\Z[\sqrt{10}]$ Consider the ring \[\Z[\sqrt{10}]=\{a+b\sqrt{10} \mid a, b \in \Z\}\] and its ideal \[P=(2, \sqrt{10})=\{a+b\sqrt{10} \mid a, b \in \Z, 2|a\}.\] Show that $p$ is a prime ideal of the ring $\Z[\sqrt{10}]$.   Definition of a prime ideal. An ideal $P$ of a ring $R$ is […]
• Nilpotent Matrix and Eigenvalues of the Matrix An $n\times n$ matrix $A$ is called nilpotent if $A^k=O$, where $O$ is the $n\times n$ zero matrix. Prove the followings. (a) The matrix $A$ is nilpotent if and only if all the eigenvalues of $A$ is zero. (b) The matrix $A$ is nilpotent if and only if […]