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Final Exam Problems and Solutions of Linear Algebra Math 2568 at the Ohio State University


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  • Describe the Range of the Matrix Using the Definition of the RangeDescribe the Range of the Matrix Using the Definition of the Range Using the definition of the range of a matrix, describe the range of the matrix \[A=\begin{bmatrix} 2 & 4 & 1 & -5 \\ 1 &2 & 1 & -2 \\ 1 & 2 & 0 & -3 \end{bmatrix}.\]   Solution. By definition, the range $\calR(A)$ of the matrix $A$ is given […]
  • Find the Eigenvalues and Eigenvectors of the Matrix $A^4-3A^3+3A^2-2A+8E$.Find the Eigenvalues and Eigenvectors of the Matrix $A^4-3A^3+3A^2-2A+8E$. Let \[A=\begin{bmatrix} 1 & -1\\ 2& 3 \end{bmatrix}.\] Find the eigenvalues and the eigenvectors of the matrix \[B=A^4-3A^3+3A^2-2A+8E.\] (Nagoya University Linear Algebra Exam Problem)   Hint. Apply the Cayley-Hamilton theorem. That is if $p_A(t)$ is the […]
  • If a Symmetric Matrix is in Reduced Row Echelon Form, then Is it Diagonal?If a Symmetric Matrix is in Reduced Row Echelon Form, then Is it Diagonal? Recall that a matrix $A$ is symmetric if $A^\trans = A$, where $A^\trans$ is the transpose of $A$. Is it true that if $A$ is a symmetric matrix and in reduced row echelon form, then $A$ is diagonal? If so, prove it. Otherwise, provide a counterexample.   Proof. […]
  • Matrix $XY-YX$ Never Be the Identity MatrixMatrix $XY-YX$ Never Be the Identity Matrix Let $I$ be the $n\times n$ identity matrix, where $n$ is a positive integer. Prove that there are no $n\times n$ matrices $X$ and $Y$ such that \[XY-YX=I.\]   Hint. Suppose that such matrices exist and consider the trace of the matrix $XY-YX$. Recall that the trace of […]
  • Centralizer, Normalizer, and Center of the Dihedral Group $D_{8}$Centralizer, Normalizer, and Center of the Dihedral Group $D_{8}$ Let $D_8$ be the dihedral group of order $8$. Using the generators and relations, we have \[D_{8}=\langle r,s \mid r^4=s^2=1, sr=r^{-1}s\rangle.\] (a) Let $A$ be the subgroup of $D_8$ generated by $r$, that is, $A=\{1,r,r^2,r^3\}$. Prove that the centralizer […]
  • Find the Rank of a Matrix with a ParameterFind the Rank of a Matrix with a Parameter Find the rank of the following real matrix. \[ \begin{bmatrix} a & 1 & 2 \\ 1 &1 &1 \\ -1 & 1 & 1-a \end{bmatrix},\] where $a$ is a real number.   (Kyoto University, Linear Algebra Exam) Solution. The rank is the number of nonzero rows of a […]
  • Ring Homomorphisms from the Ring of Rational Numbers are Determined by the Values at IntegersRing Homomorphisms from the Ring of Rational Numbers are Determined by the Values at Integers Let $R$ be a ring with unity. Suppose that $f$ and $g$ are ring homomorphisms from $\Q$ to $R$ such that $f(n)=g(n)$ for any integer $n$. Then prove that $f=g$.   Proof. Let $a/b \in \Q$ be an arbitrary rational number with integers $a, b$. Then we […]
  • True or False Problems on Midterm Exam 1 at OSU Spring 2018True or False Problems on Midterm Exam 1 at OSU Spring 2018 The following problems are True or False. Let $A$ and $B$ be $n\times n$ matrices. (a) If $AB=B$, then $B$ is the identity matrix. (b) If the coefficient matrix $A$ of the system $A\mathbf{x}=\mathbf{b}$ is invertible, then the system has infinitely many solutions. (c) If $A$ […]

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