# symmetric-matrix

• Determine Whether Given Subsets in $\R^4$ are Subspaces or Not (a) Let $S$ be the subset of $\R^4$ consisting of vectors $\begin{bmatrix} x \\ y \\ z \\ w \end{bmatrix}$ satisfying $2x+4y+3z+7w+1=0.$ Determine whether $S$ is a subspace of $\R^4$. If so prove it. If not, explain why it is not a […]
• Find a Condition that a Vector be a Linear Combination Let $\mathbf{v}=\begin{bmatrix} a \\ b \\ c \end{bmatrix}, \qquad \mathbf{v}_1=\begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix}, \qquad \mathbf{v}_2=\begin{bmatrix} 2 \\ -1 \\ 2 \end{bmatrix}.$ Find the necessary and […]
• Find All Symmetric Matrices satisfying the Equation Find all $2\times 2$ symmetric matrices $A$ satisfying $A\begin{bmatrix} 1 \\ -1 \end{bmatrix} = \begin{bmatrix} 2 \\ 3 \end{bmatrix}$? Express your solution using free variable(s).   Solution. Let $A=\begin{bmatrix} a & b\\ c& d \end{bmatrix}$ […]
• Automorphism Group of $\Q(\sqrt[3]{2})$ Over $\Q$. Determine the automorphism group of $\Q(\sqrt[3]{2})$ over $\Q$. Proof. Let $\sigma \in \Aut(\Q(\sqrt[3]{2}/\Q)$ be an automorphism of $\Q(\sqrt[3]{2})$ over $\Q$. Then $\sigma$ is determined by the value $\sigma(\sqrt[3]{2})$ since any element $\alpha$ of $\Q(\sqrt[3]{2})$ […]
• If a Matrix $A$ is Full Rank, then $\rref(A)$ is the Identity Matrix Prove that if $A$ is an $n \times n$ matrix with rank $n$, then $\rref(A)$ is the identity matrix. Here $\rref(A)$ is the matrix in reduced row echelon form that is row equivalent to the matrix $A$.   Proof. Because $A$ has rank $n$, we know that the $n \times n$ […]
• Solve the System of Linear Equations Using the Inverse Matrix of the Coefficient Matrix Consider the following system of linear equations \begin{align*} 2x+3y+z&=-1\\ 3x+3y+z&=1\\ 2x+4y+z&=-2. \end{align*} (a) Find the coefficient matrix $A$ for this system. (b) Find the inverse matrix of the coefficient matrix found in (a) (c) Solve the system using […]
• A Singular Matrix and Matrix Equations $A\mathbf{x}=\mathbf{e}_i$ With Unit Vectors Let $A$ be a singular $n\times n$ matrix. Let \[\mathbf{e}_1=\begin{bmatrix} 1 \\ 0 \\ \vdots \\ 0 \end{bmatrix}, \mathbf{e}_2=\begin{bmatrix} 0 \\ 1 \\ \vdots \\ 0 \end{bmatrix}, \dots, […]
• The Inverse Matrix is Unique Let $A$ be an $n\times n$ invertible matrix. Prove that the inverse matrix of $A$ is uniques.   Hint. That the inverse matrix of $A$ is unique means that there is only one inverse matrix of $A$. (That's why we say "the" inverse matrix of $A$ and denote it by […]