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• The Product of Two Nonsingular Matrices is Nonsingular Prove that if $n\times n$ matrices $A$ and $B$ are nonsingular, then the product $AB$ is also a nonsingular matrix. (The Ohio State University, Linear Algebra Final Exam Problem)   Definition (Nonsingular Matrix) An $n\times n$ matrix is called nonsingular if the […]
• Is the Quotient Ring of an Integral Domain still an Integral Domain? Let $R$ be an integral domain and let $I$ be an ideal of $R$. Is the quotient ring $R/I$ an integral domain?   Definition (Integral Domain). Let $R$ be a commutative ring. An element $a$ in $R$ is called a zero divisor if there exists $b\neq 0$ in $R$ such that […]
• Group of Order $pq$ is Either Abelian or the Center is Trivial Let $G$ be a group of order $|G|=pq$, where $p$ and $q$ are (not necessarily distinct) prime numbers. Then show that $G$ is either abelian group or the center $Z(G)=1$. Hint. Use the result of the problem "If the Quotient by the Center is Cyclic, then the Group is […]
• Determine When the Given Matrix Invertible For which choice(s) of the constant $k$ is the following matrix invertible? \[A=\begin{bmatrix} 1 & 1 & 1 \\ 1 &2 &k \\ 1 & 4 & k^2 \end{bmatrix}.\]   (Johns Hopkins University, Linear Algebra Exam)   Hint. An $n\times n$ matrix is […]
• No/Infinitely Many Square Roots of 2 by 2 Matrices (a) Prove that the matrix $A=\begin{bmatrix} 0 & 1\\ 0& 0 \end{bmatrix}$ does not have a square root. Namely, show that there is no complex matrix $B$ such that $B^2=A$. (b) Prove that the $2\times 2$ identity matrix $I$ has infinitely many distinct square root […]
• If $R$ is a Noetherian Ring and $f:R\to R’$ is a Surjective Homomorphism, then $R’$ is Noetherian Suppose that $f:R\to R'$ is a surjective ring homomorphism. Prove that if $R$ is a Noetherian ring, then so is $R'$.   Definition. A ring $S$ is Noetherian if for every ascending chain of ideals of $S$ \[I_1 \subset I_2 \subset \cdots \subset I_k \subset […]
• If Two Ideals Are Comaximal in a Commutative Ring, then Their Powers Are Comaximal Ideals Let $R$ be a commutative ring and let $I_1$ and $I_2$ be comaximal ideals. That is, we have \[I_1+I_2=R.\] Then show that for any positive integers $m$ and $n$, the ideals $I_1^m$ and $I_2^n$ are comaximal.   > Proof. Since $I_1+I_2=R$, there exists $a \in I_1$ […]
• Determine whether the Given 3 by 3 Matrices are Nonsingular Determine whether the following matrices are nonsingular or not. (a) $A=\begin{bmatrix} 1 & 0 & 1 \\ 2 &1 &2 \\ 1 & 0 & -1 \end{bmatrix}$. (b) $B=\begin{bmatrix} 2 & 1 & 2 \\ 1 &0 &1 \\ 4 & 1 & 4 \end{bmatrix}$.   Solution. Recall that […]