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• Solving a System of Linear Equations By Using an Inverse Matrix Consider the system of linear equations \begin{align*} x_1&= 2, \\ -2x_1 + x_2 &= 3, \\ 5x_1-4x_2 +x_3 &= 2 \end{align*} (a) Find the coefficient matrix and its inverse matrix. (b) Using the inverse matrix, solve the system of linear equations. (The Ohio […]
• How to Prove a Matrix is Nonsingular in 10 Seconds Using the numbers appearing in $\pi=3.1415926535897932384626433832795028841971693993751058209749\dots$ we construct the matrix $A=\begin{bmatrix} 3 & 14 &1592& 65358\\ 97932& 38462643& 38& 32\\ 7950& 2& 8841& 9716\\ 939937510& 5820& 974& […] • Eigenvalues and their Algebraic Multiplicities of a Matrix with a Variable Determine all eigenvalues and their algebraic multiplicities of the matrix \[A=\begin{bmatrix} 1 & a & 1 \\ a &1 &a \\ 1 & a & 1 \end{bmatrix},$ where $a$ is a real number.   Proof. To find eigenvalues we first compute the characteristic polynomial of the […]
• Powers of a Diagonal Matrix Let $A=\begin{bmatrix} a & 0\\ 0& b \end{bmatrix}$. Show that (1) $A^n=\begin{bmatrix} a^n & 0\\ 0& b^n \end{bmatrix}$ for any $n \in \N$. (2) Let $B=S^{-1}AS$, where $S$ be an invertible $2 \times 2$ matrix. Show that $B^n=S^{-1}A^n S$ for any $n \in […] • True or False Problems of Vector Spaces and Linear Transformations These are True or False problems. For each of the following statements, determine if it contains a wrong information or not. Let$A$be a$5\times 3$matrix. Then the range of$A$is a subspace in$\R^3$. The function$f(x)=x^2+1$is not in the vector space$C[-1,1]$because […] • Given Eigenvectors and Eigenvalues, Compute a Matrix Product (Stanford University Exam) Suppose that$\begin{bmatrix} 1 \\ 1 \end{bmatrix}$is an eigenvector of a matrix$A$corresponding to the eigenvalue$3$and that$\begin{bmatrix} 2 \\ 1 \end{bmatrix}$is an eigenvector of$A$corresponding to the eigenvalue$-2$. Compute$A^2\begin{bmatrix} 4 […]
• Subgroup of Finite Index Contains a Normal Subgroup of Finite Index Let $G$ be a group and let $H$ be a subgroup of finite index. Then show that there exists a normal subgroup $N$ of $G$ such that $N$ is of finite index in $G$ and $N\subset H$.   Proof. The group $G$ acts on the set of left cosets $G/H$ by left multiplication. Hence […]
• Restriction of a Linear Transformation on the x-z Plane is a Linear Transformation Let $T:\R^3 \to \R^3$ be a linear transformation and suppose that its matrix representation with respect to the standard basis is given by the matrix $A=\begin{bmatrix} 1 & 0 & 2 \\ 0 &3 &0 \\ 4 & 0 & 5 \end{bmatrix}.$ (a) Prove that the linear transformation […]