Union of Two Subgroups is Not a Group
Let $G$ be a group and let $H_1, H_2$ be subgroups of $G$ such that $H_1 \not \subset H_2$ and $H_2 \not \subset H_1$.
(a) Prove that the union $H_1 \cup H_2$ is never a subgroup in $G$.
(b) Prove that a group cannot be written as the union of two proper […]
For What Values of $a$, Is the Matrix Nonsingular?
Determine the values of a real number $a$ such that the matrix
\[A=\begin{bmatrix}
3 & 0 & a \\
2 &3 &0 \\
0 & 18a & a+1
\end{bmatrix}\]
is nonsingular.
Solution.
We apply elementary row operations and obtain:
\begin{align*}
A=\begin{bmatrix}
3 & 0 & a […]
If the Localization is Noetherian for All Prime Ideals, Is the Ring Noetherian?
Let $R$ be a commutative ring with $1$.
Suppose that the localization $R_{\mathfrak{p}}$ is a Noetherian ring for every prime ideal $\mathfrak{p}$ of $R$.
Is it true that $A$ is also a Noetherian ring?
Proof.
The answer is no. We give a counterexample.
Let […]
Example of an Infinite Group Whose Elements Have Finite Orders
Is it possible that each element of an infinite group has a finite order?
If so, give an example. Otherwise, prove the non-existence of such a group.
Solution.
We give an example of a group of infinite order each of whose elements has a finite order.
Consider […]
In a Field of Positive Characteristic, $A^p=I$ Does Not Imply that $A$ is Diagonalizable.
Show that the matrix $A=\begin{bmatrix}
1 & \alpha\\
0& 1
\end{bmatrix}$, where $\alpha$ is an element of a field $F$ of characteristic $p>0$ satisfies $A^p=I$ and the matrix is not diagonalizable over $F$ if $\alpha \neq 0$.
Comment.
Remark that if $A$ is a square […]
Is the Trace of the Transposed Matrix the Same as the Trace of the Matrix?
Let $A$ be an $n \times n$ matrix.
Is it true that $\tr ( A^\trans ) = \tr(A)$? If it is true, prove it. If not, give a counterexample.
Solution.
The answer is true. Recall that the transpose of a matrix is the sum of its diagonal entries. Also, note that the […]
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Union of Two Subgroups is Not a Group
Let $G$ be a group and let $H_1, H_2$ be subgroups of $G$ such that $H_1 \not \subset H_2$ and $H_2 \not \subset H_1$.
(a) Prove that the union $H_1 \cup H_2$ is never a subgroup in $G$.
(b) Prove that a group cannot be written as the union of two proper […]
For What Values of $a$, Is the Matrix Nonsingular?
Determine the values of a real number $a$ such that the matrix
\[A=\begin{bmatrix}
3 & 0 & a \\
2 &3 &0 \\
0 & 18a & a+1
\end{bmatrix}\]
is nonsingular.
Solution.
We apply elementary row operations and obtain:
\begin{align*}
A=\begin{bmatrix}
3 & 0 & a […]
If the Localization is Noetherian for All Prime Ideals, Is the Ring Noetherian?
Let $R$ be a commutative ring with $1$.
Suppose that the localization $R_{\mathfrak{p}}$ is a Noetherian ring for every prime ideal $\mathfrak{p}$ of $R$.
Is it true that $A$ is also a Noetherian ring?
Proof.
The answer is no. We give a counterexample.
Let […]
Example of an Infinite Group Whose Elements Have Finite Orders
Is it possible that each element of an infinite group has a finite order?
If so, give an example. Otherwise, prove the non-existence of such a group.
Solution.
We give an example of a group of infinite order each of whose elements has a finite order.
Consider […]
Possibilities of the Number of Solutions of a Homogeneous System of Linear Equations
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