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• Find the Inverse Matrix of a Matrix With Fractions Find the inverse matrix of the matrix \[A=\begin{bmatrix} \frac{2}{7} & \frac{3}{7} & \frac{6}{7} \\[6 pt] \frac{6}{7} &\frac{2}{7} &-\frac{3}{7} \\[6pt] -\frac{3}{7} & \frac{6}{7} & -\frac{2}{7} \end{bmatrix}.\]   Hint. You may use the augmented matrix […]
• Basis and Dimension of the Subspace of All Polynomials of Degree 4 or Less Satisfying Some Conditions. Let $P_4$ be the vector space consisting of all polynomials of degree $4$ or less with real number coefficients. Let $W$ be the subspace of $P_2$ by \[W=\{ p(x)\in P_4 \mid p(1)+p(-1)=0 \text{ and } p(2)+p(-2)=0 \}.\] Find a basis of the subspace $W$ and determine the dimension of […]
• If Squares of Elements in a Group Lie in a Subgroup, then It is a Normal Subgroup Let $H$ be a subgroup of a group $G$. Suppose that for each element $x\in G$, we have $x^2\in H$. Then prove that $H$ is a normal subgroup of $G$. (Purdue University, Abstract Algebra Qualifying Exam)   Proof. To show that $H$ is a normal subgroup of […]
• A One Side Inverse Matrix is the Inverse Matrix: If $AB=I$, then $BA=I$ An $n\times n$ matrix $A$ is said to be invertible if there exists an $n\times n$ matrix $B$ such that $AB=I$, and $BA=I$, where $I$ is the $n\times n$ identity matrix. If such a matrix $B$ exists, then it is known to be unique and called the inverse matrix of $A$, denoted […]
• If Quotient $G/H$ is Abelian Group and $H < K \triangleleft　G$, then $G/K$ is Abelian Let $H$ and $K$ be normal subgroups of a group $G$. Suppose that $H < K$ and the quotient group $G/H$ is abelian. Then prove that $G/K$ is also an abelian group.   Solution. We will give two proofs. Hint (The third isomorphism theorem) Recall the third […]
• Possibilities of the Number of Solutions of a Homogeneous System of Linear Equations Here is a very short true or false problem. Select either True or False. Then click "Finish quiz" button. You will be able to see an explanation of the solution by clicking "View questions" button.  
• Sylow Subgroups of a Group of Order 33 is Normal Subgroups Prove that any $p$-Sylow subgroup of a group $G$ of order $33$ is a normal subgroup of $G$.   Hint. We use Sylow's theorem. Review the basic terminologies and Sylow's theorem. Recall that if there is only one $p$-Sylow subgroup $P$ of $G$ for a fixed prime $p$, then $P$ […]
• All Linear Transformations that Take the Line $y=x$ to the Line $y=-x$ Determine all linear transformations of the $2$-dimensional $x$-$y$ plane $\R^2$ that take the line $y=x$ to the line $y=-x$.   Solution. Let $T:\R^2 \to \R^2$ be a linear transformation that maps the line $y=x$ to the line $y=-x$. Note that the linear […]