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  • If the Matrix Product $AB=0$, then is $BA=0$ as Well?If the Matrix Product $AB=0$, then is $BA=0$ as Well? Let $A$ and $B$ be $n\times n$ matrices. Suppose that the matrix product $AB=O$, where $O$ is the $n\times n$ zero matrix. Is it true that the matrix product with opposite order $BA$ is also the zero matrix? If so, give a proof. If not, give a […]
  • The Inner Product on $\R^2$ induced by a Positive Definite Matrix and Gram-Schmidt OrthogonalizationThe Inner Product on $\R^2$ induced by a Positive Definite Matrix and Gram-Schmidt Orthogonalization Consider the $2\times 2$ real matrix \[A=\begin{bmatrix} 1 & 1\\ 1& 3 \end{bmatrix}.\] (a) Prove that the matrix $A$ is positive definite. (b) Since $A$ is positive definite by part (a), the formula \[\langle \mathbf{x}, […]
  • Row Equivalent Matrix, Bases for the Null Space, Range, and Row Space of a MatrixRow Equivalent Matrix, Bases for the Null Space, Range, and Row Space of a Matrix Let \[A=\begin{bmatrix} 1 & 1 & 2 \\ 2 &2 &4 \\ 2 & 3 & 5 \end{bmatrix}.\] (a) Find a matrix $B$ in reduced row echelon form such that $B$ is row equivalent to the matrix $A$. (b) Find a basis for the null space of $A$. (c) Find a basis for the range of $A$ that […]
  • Is there an Odd Matrix Whose Square is $-I$?Is there an Odd Matrix Whose Square is $-I$? Let $n$ be an odd positive integer. Determine whether there exists an $n \times n$ real matrix $A$ such that \[A^2+I=O,\] where $I$ is the $n \times n$ identity matrix and $O$ is the $n \times n$ zero matrix. If such a matrix $A$ exists, find an example. If not, prove that […]
  • Summary: Possibilities for the Solution Set of a System of Linear EquationsSummary: Possibilities for the Solution Set of a System of Linear Equations In this post, we summarize theorems about the possibilities for the solution set of a system of linear equations and solve the following problems. Determine all possibilities for the solution set of the system of linear equations described below. (a) A homogeneous system of $3$ […]
  • Irreducible Polynomial $x^3+9x+6$ and Inverse Element in Field ExtensionIrreducible Polynomial $x^3+9x+6$ and Inverse Element in Field Extension Prove that the polynomial \[f(x)=x^3+9x+6\] is irreducible over the field of rational numbers $\Q$. Let $\theta$ be a root of $f(x)$. Then find the inverse of $1+\theta$ in the field $\Q(\theta)$.   Proof. Note that $f(x)$ is a monic polynomial and the prime […]
  • Matrices Satisfying the Relation $HE-EH=2E$Matrices Satisfying the Relation $HE-EH=2E$ Let $H$ and $E$ be $n \times n$ matrices satisfying the relation \[HE-EH=2E.\] Let $\lambda$ be an eigenvalue of the matrix $H$ such that the real part of $\lambda$ is the largest among the eigenvalues of $H$. Let $\mathbf{x}$ be an eigenvector corresponding to $\lambda$. Then […]
  • The Centralizer of a Matrix is a SubspaceThe Centralizer of a Matrix is a Subspace Let $V$ be the vector space of $n \times n$ matrices, and $M \in V$ a fixed matrix. Define \[W = \{ A \in V \mid AM = MA \}.\] The set $W$ here is called the centralizer of $M$ in $V$. Prove that $W$ is a subspace of $V$.   Proof. First we check that the zero […]

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