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Field theory problems and solution in abstract algebra


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  • True of False Problems on Determinants and Invertible MatricesTrue of False Problems on Determinants and Invertible Matrices Determine whether each of the following statements is True or False. (a) If $A$ and $B$ are $n \times n$ matrices, and $P$ is an invertible $n \times n$ matrix such that $A=PBP^{-1}$, then $\det(A)=\det(B)$. (b) If the characteristic polynomial of an $n \times n$ matrix $A$ […]
  • Even Perfect Numbers and Mersenne Prime NumbersEven Perfect Numbers and Mersenne Prime Numbers Prove that if $2^n-1$ is a Mersenne prime number, then \[N=2^{n-1}(2^n-1)\] is a perfect number. On the other hand, prove that every even perfect number $N$ can be written as $N=2^{n-1}(2^n-1)$ for some Mersenne prime number $2^n-1$.   Definitions. In this post, a […]
  • Give the Formula for a Linear Transformation from $\R^3$ to $\R^2$Give the Formula for a Linear Transformation from $\R^3$ to $\R^2$ Let $T: \R^3 \to \R^2$ be a linear transformation such that \[T(\mathbf{e}_1)=\begin{bmatrix} 1 \\ 4 \end{bmatrix}, T(\mathbf{e}_2)=\begin{bmatrix} 2 \\ 5 \end{bmatrix}, T(\mathbf{e}_3)=\begin{bmatrix} 3 \\ 6 […]
  • Diagonalize the 3 by 3 Matrix if it is DiagonalizableDiagonalize the 3 by 3 Matrix if it is Diagonalizable Determine whether the matrix \[A=\begin{bmatrix} 0 & 1 & 0 \\ -1 &0 &0 \\ 0 & 0 & 2 \end{bmatrix}\] is diagonalizable. If it is diagonalizable, then find the invertible matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.   How to […]
  • Group Homomorphism, Preimage, and Product of GroupsGroup Homomorphism, Preimage, and Product of Groups Let $G, G'$ be groups and let $f:G \to G'$ be a group homomorphism. Put $N=\ker(f)$. Then show that we have \[f^{-1}(f(H))=HN.\]   Proof. $(\subset)$ Take an arbitrary element $g\in f^{-1}(f(H))$. Then we have $f(g)\in f(H)$. It follows that there exists $h\in H$ […]
  • Determine a Matrix From Its EigenvalueDetermine a Matrix From Its Eigenvalue Let \[A=\begin{bmatrix} a & -1\\ 1& 4 \end{bmatrix}\] be a $2\times 2$ matrix, where $a$ is some real number. Suppose that the matrix $A$ has an eigenvalue $3$. (a) Determine the value of $a$. (b) Does the matrix $A$ have eigenvalues other than […]
  • Normal Subgroups Intersecting Trivially Commute in a GroupNormal Subgroups Intersecting Trivially Commute in a Group Let $A$ and $B$ be normal subgroups of a group $G$. Suppose $A\cap B=\{e\}$, where $e$ is the unit element of the group $G$. Show that for any $a \in A$ and $b \in B$ we have $ab=ba$. Hint. Consider the commutator of $a$ and $b$, that […]
  • How to Find the Determinant of the  $3\times 3$ MatrixHow to Find the Determinant of the $3\times 3$ Matrix Find the determinant of the matix \[A=\begin{bmatrix} 100 & 101 & 102 \\ 101 &102 &103 \\ 102 & 103 & 104 \end{bmatrix}.\]   Solution. Note that the determinant does not change if the $i$-th row is added by a scalar multiple of the $j$-th row if $i \neq […]

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