# Michigan-State-University-Abstract-Algebra-eye-catch

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• Find Values of $a, b, c$ such that the Given Matrix is Diagonalizable For which values of constants $a, b$ and $c$ is the matrix $A=\begin{bmatrix} 7 & a & b \\ 0 &2 &c \\ 0 & 0 & 3 \end{bmatrix}$ diagonalizable? (The Ohio State University, Linear Algebra Final Exam Problem)   Solution. Note that the […]
• Powers of a Matrix Cannot be a Basis of the Vector Space of Matrices Let $n>1$ be a positive integer. Let $V=M_{n\times n}(\C)$ be the vector space over the complex numbers $\C$ consisting of all complex $n\times n$ matrices. The dimension of $V$ is $n^2$. Let $A \in V$ and consider the set $S_A=\{I=A^0, A, A^2, \dots, A^{n^2-1}\}$ of $n^2$ […]
• Linear Algebra Midterm 1 at the Ohio State University (2/3) The following problems are Midterm 1 problems of Linear Algebra (Math 2568) at the Ohio State University in Autumn 2017. There were 9 problems that covered Chapter 1 of our textbook (Johnson, Riess, Arnold). The time limit was 55 minutes. This post is Part 2 and contains […]
• Find All Values of $x$ so that a Matrix is Singular Let $A=\begin{bmatrix} 1 & -x & 0 & 0 \\ 0 &1 & -x & 0 \\ 0 & 0 & 1 & -x \\ 0 & 1 & 0 & -1 \end{bmatrix}$ be a $4\times 4$ matrix. Find all values of $x$ so that the matrix $A$ is singular.   Hint. Use the fact that a matrix is singular if and only […]
• Find 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 […]
• A Line is a Subspace if and only if its $y$-Intercept is Zero Let $\R^2$ be the $x$-$y$-plane. Then $\R^2$ is a vector space. A line $\ell \subset \mathbb{R}^2$ with slope $m$ and $y$-intercept $b$ is defined by $\ell = \{ (x, y) \in \mathbb{R}^2 \mid y = mx + b \} .$ Prove that $\ell$ is a subspace of $\mathbb{R}^2$ if and only if $b = […] • If Column Vectors Form Orthonormal set, is Row Vectors Form Orthonormal Set? Suppose that$A$is a real$n\times n$matrix. (a) Is it true that$A$must commute with its transpose? (b) Suppose that the columns of$A$(considered as vectors) form an orthonormal set. Is it true that the rows of$A$must also form an orthonormal set? (University of […] • Every Group of Order 20449 is an Abelian Group Prove that every group of order$20449$is an abelian group. Outline of the Proof Note that$20449=11^2 \cdot 13^2$. Let$G$be a group of order$20449$. We prove by Sylow's theorem that there are a unique Sylow$11$-subgroup and a unique Sylow$13\$-subgroup of […]