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• 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 […]
• Determine Whether There Exists a Nonsingular Matrix Satisfying $A^4=ABA^2+2A^3$ Determine whether there exists a nonsingular matrix $A$ if $A^4=ABA^2+2A^3,$ where $B$ is the following matrix. $B=\begin{bmatrix} -1 & 1 & -1 \\ 0 &-1 &0 \\ 2 & 1 & -4 \end{bmatrix}.$ If such a nonsingular matrix $A$ exists, find the inverse […]
• Inequality about Eigenvalue of a Real Symmetric Matrix Let $A$ be an $n\times n$ real symmetric matrix. Prove that there exists an eigenvalue $\lambda$ of $A$ such that for any vector $\mathbf{v}\in \R^n$, we have the inequality $\mathbf{v}\cdot A\mathbf{v} \leq \lambda \|\mathbf{v}\|^2.$     Proof. Recall […]
• A Diagonalizable Matrix which is Not Diagonalized by a Real Nonsingular Matrix Prove that the matrix $A=\begin{bmatrix} 0 & 1\\ -1& 0 \end{bmatrix}$ is diagonalizable. Prove, however, that $A$ cannot be diagonalized by a real nonsingular matrix. That is, there is no real nonsingular matrix $S$ such that $S^{-1}AS$ is a diagonal […]
• Are Groups of Order 100, 200 Simple? Determine whether a group $G$ of the following order is simple or not. (a) $|G|=100$. (b) $|G|=200$.   Hint. Use Sylow's theorem and determine the number of $5$-Sylow subgroup of the group $G$. Check out the post Sylow’s Theorem (summary) for a review of Sylow's […]
• Ring Homomorphisms and Radical Ideals Let $R$ and $R'$ be commutative rings and let $f:R\to R'$ be a ring homomorphism. Let $I$ and $I'$ be ideals of $R$ and $R'$, respectively. (a) Prove that $f(\sqrt{I}\,) \subset \sqrt{f(I)}$. (b) Prove that $\sqrt{f^{-1}(I')}=f^{-1}(\sqrt{I'})$ (c) Suppose that $f$ is […]
• Find the Inverse Matrices if Matrices are Invertible by Elementary Row Operations For each of the following $3\times 3$ matrices $A$, determine whether $A$ is invertible and find the inverse $A^{-1}$ if exists by computing the augmented matrix $[A|I]$, where $I$ is the $3\times 3$ identity matrix. (a) $A=\begin{bmatrix} 1 & 3 & -2 \\ 2 &3 &0 \\ […] • Determinant/Trace and Eigenvalues of a Matrix Let$A$be an$n\times n$matrix and let$\lambda_1, \dots, \lambda_n$be its eigenvalues. Show that (1) $$\det(A)=\prod_{i=1}^n \lambda_i$$ (2) $$\tr(A)=\sum_{i=1}^n \lambda_i$$ Here$\det(A)$is the determinant of the matrix$A$and$\tr(A)\$ is the trace of the matrix […]