More from my site

• Linear Combination of Eigenvectors is Not an Eigenvector Suppose that $\lambda$ and $\mu$ are two distinct eigenvalues of a square matrix $A$ and let $\mathbf{x}$ and $\mathbf{y}$ be eigenvectors corresponding to $\lambda$ and $\mu$, respectively. If $a$ and $b$ are nonzero numbers, then prove that $a \mathbf{x}+b\mathbf{y}$ is not an […]
• The Powers of the Matrix with Cosine and Sine Functions Prove the following identity for any positive integer $n$. \[\begin{bmatrix} \cos \theta & -\sin \theta\\ \sin \theta& \cos \theta \end{bmatrix}^n=\begin{bmatrix} \cos n\theta & -\sin n\theta\\ \sin n\theta& \cos […]
• Prove a Group is Abelian if $(ab)^3=a^3b^3$ and No Elements of Order $3$ Let $G$ be a group. Suppose that we have \[(ab)^3=a^3b^3\] for any elements $a, b$ in $G$. Also suppose that $G$ has no elements of order $3$. Then prove that $G$ is an abelian group.   Proof. Let $a, b$ be arbitrary elements of the group $G$. We want […]
• If Two Matrices are Similar, then their Determinants are the Same Prove that if $A$ and $B$ are similar matrices, then their determinants are the same.   Proof. Suppose that $A$ and $B$ are similar. Then there exists a nonsingular matrix $S$ such that \[S^{-1}AS=B\] by definition. Then we […]
• Group Homomorphisms From Group of Order 21 to Group of Order 49 Let $G$ be a finite group of order $21$ and let $K$ be a finite group of order $49$. Suppose that $G$ does not have a normal subgroup of order $3$. Then determine all group homomorphisms from $G$ to $K$.   Proof. Let $e$ be the identity element of the group […]
• Sequences Satisfying Linear Recurrence Relation Form a Subspace Let $V$ be a real vector space of all real sequences \[(a_i)_{i=1}^{\infty}=(a_1, a_2, \cdots).\] Let $U$ be the subset of $V$ defined by \[U=\{ (a_i)_{i=1}^{\infty} \in V \mid a_{k+2}-5a_{k+1}+3a_{k}=0, k=1, 2, \dots \}.\] Prove that $U$ is a subspace of […]
• Is a Set of All Nilpotent Matrix a Vector Space? Let $V$ denote the vector space of all real $n\times n$ matrices, where $n$ is a positive integer. Determine whether the set $U$ of all $n\times n$ nilpotent matrices is a subspace of the vector space $V$ or not.   Definition. An matrix $A$ is a nilpotent matrix if […]
• The Determinant of a Skew-Symmetric Matrix is Zero Prove that the determinant of an $n\times n$ skew-symmetric matrix is zero if $n$ is odd.   Definition (Skew-Symmetric) A matrix $A$ is called skew-symmetric if $A^{\trans}=-A$. Here $A^{\trans}$ is the transpose of $A$. Proof. Properties of […]