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  • Eigenvalues of Squared Matrix and Upper Triangular MatrixEigenvalues of Squared Matrix and Upper Triangular Matrix Suppose that $A$ and $P$ are $3 \times 3$ matrices and $P$ is invertible matrix. If \[P^{-1}AP=\begin{bmatrix} 1 & 2 & 3 \\ 0 &4 &5 \\ 0 & 0 & 6 \end{bmatrix},\] then find all the eigenvalues of the matrix $A^2$.   We give two proofs. The first version is a […]
  • Find a Quadratic Function Satisfying Conditions on DerivativesFind a Quadratic Function Satisfying Conditions on Derivatives Find a quadratic function $f(x) = ax^2 + bx + c$ such that $f(1) = 3$, $f'(1) = 3$, and $f^{\prime\prime}(1) = 2$. Here, $f'(x)$ and $f^{\prime\prime}(x)$ denote the first and second derivatives, respectively.   Solution. Each condition required on $f$ can be turned […]
  • If a Prime Ideal Contains No Nonzero Zero Divisors, then the Ring is an Integral DomainIf a Prime Ideal Contains No Nonzero Zero Divisors, then the Ring is an Integral Domain Let $R$ be a commutative ring. Suppose that $P$ is a prime ideal of $R$ containing no nonzero zero divisor. Then show that the ring $R$ is an integral domain.   Definitions: zero divisor, integral domain An element $a$ of a commutative ring $R$ is called a zero divisor […]
  • Diagonalize a 2 by 2 Matrix if DiagonalizableDiagonalize a 2 by 2 Matrix if Diagonalizable Determine whether the matrix \[A=\begin{bmatrix} 1 & 4\\ 2 & 3 \end{bmatrix}\] is diagonalizable. If so, find a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$. (The Ohio State University, Linear Algebra Final Exam […]
  • The Image of an Ideal Under a Surjective Ring Homomorphism is an IdealThe Image of an Ideal Under a Surjective Ring Homomorphism is an Ideal Let $R$ and $S$ be rings. Suppose that $f: R \to S$ is a surjective ring homomorphism. Prove that every image of an ideal of $R$ under $f$ is an ideal of $S$. Namely, prove that if $I$ is an ideal of $R$, then $J=f(I)$ is an ideal of $S$.   Proof. As in the […]
  • Top 10 Popular Math Problems in 2016-2017Top 10 Popular Math Problems in 2016-2017 It's been a year since I started this math blog!! More than 500 problems were posted during a year (July 19th 2016-July 19th 2017). I made a list of the 10 math problems on this blog that have the most views. Can you solve all of them? The level of difficulty among the top […]
  • Questions About the Trace of a MatrixQuestions About the Trace of a Matrix Let $A=(a_{i j})$ and $B=(b_{i j})$ be $n\times n$ real matrices for some $n \in \N$. Then answer the following questions about the trace of a matrix. (a) Express $\tr(AB^{\trans})$ in terms of the entries of the matrices $A$ and $B$. Here $B^{\trans}$ is the transpose matrix of […]
  • Solve the System of Linear Equations and Give the Vector Form for the General SolutionSolve the System of Linear Equations and Give the Vector Form for the General Solution Solve the following system of linear equations and give the vector form for the general solution. \begin{align*} x_1 -x_3 -2x_5&=1 \\ x_2+3x_3-x_5 &=2 \\ 2x_1 -2x_3 +x_4 -3x_5 &= 0 \end{align*} (The Ohio State University, linear algebra midterm exam […]

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