# find-eigenvalues-eigenvectors

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• Is there an Odd Matrix Whose Square is $-I$? Let $n$ be an odd positive integer. Determine whether there exists an $n \times n$ real matrix $A$ such that $A^2+I=O,$ where $I$ is the $n \times n$ identity matrix and $O$ is the $n \times n$ zero matrix. If such a matrix $A$ exists, find an example. If not, prove that […]
• True or False. Every Diagonalizable Matrix is Invertible Is every diagonalizable matrix invertible?   Solution. The answer is No. Counterexample We give a counterexample. Consider the $2\times 2$ zero matrix. The zero matrix is a diagonal matrix, and thus it is diagonalizable. However, the zero matrix is not […]
• 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 […]
• Find the Inverse Matrices if Matrices are Invertible by Elementary Row Operations For each of the following $3\times 3$ matrices $A$, determine whether $A$ is invertible and find the inverse $A^{-1}$ if exists by computing the augmented matrix $[A|I]$, where $I$ is the $3\times 3$ identity matrix. (a) $A=\begin{bmatrix} 1 & 3 & -2 \\ 2 &3 &0 \\ […] • Group Homomorphisms From Group of Order 21 to Group of Order 49 Let$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 […] • The Cyclotomic Field of 8-th Roots of Unity is$\Q(\zeta_8)=\Q(i, \sqrt{2})$Let$\zeta_8$be a primitive$8$-th root of unity. Prove that the cyclotomic field$\Q(\zeta_8)$of the$8$-th root of unity is the field$\Q(i, \sqrt{2})$. Proof. Recall that the extension degree of the cyclotomic field of$n$-th roots of unity is given by […] • 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 […] • Linear Transformation and a Basis of the Vector Space$\R^3$Let$T$be a linear transformation from the vector space$\R^3$to$\R^3$. Suppose that$k=3$is the smallest positive integer such that$T^k=\mathbf{0}$(the zero linear transformation) and suppose that we have$\mathbf{x}\in \R^3$such that$T^2\mathbf{x}\neq \mathbf{0}\$. Show […]