# graphs-of-characteristic-polynomials

• Are these vectors in the Nullspace of the Matrix? Let $A=\begin{bmatrix} 1 & 0 & 3 & -2 \\ 0 &3 & 1 & 1 \\ 1 & 3 & 4 & -1 \end{bmatrix}$. For each of the following vectors, determine whether the vector is in the nullspace $\calN(A)$. (a) $\begin{bmatrix} -3 \\ 0 \\ 1 \\ 0 \end{bmatrix}$ […]
• Inverse Matrix Contains Only Integers if and only if the Determinant is $\pm 1$ Let $A$ be an $n\times n$ nonsingular matrix with integer entries. Prove that the inverse matrix $A^{-1}$ contains only integer entries if and only if $\det(A)=\pm 1$.   Hint. If $B$ is a square matrix whose entries are integers, then the […]
• Rank and Nullity of Linear Transformation From $\R^3$ to $\R^2$ Let $T:\R^3 \to \R^2$ be a linear transformation such that $T(\mathbf{e}_1)=\begin{bmatrix} 1 \\ 0 \end{bmatrix}, T(\mathbf{e}_2)=\begin{bmatrix} 0 \\ 1 \end{bmatrix}, T(\mathbf{e}_3)=\begin{bmatrix} 1 \\ 0 \end{bmatrix},$ where $\mathbf{e}_1, […] • Positive definite Real Symmetric Matrix and its Eigenvalues A real symmetric$n \times n$matrix$A$is called positive definite if $\mathbf{x}^{\trans}A\mathbf{x}>0$ for all nonzero vectors$\mathbf{x}$in$\R^n$. (a) Prove that the eigenvalues of a real symmetric positive-definite matrix$A$are all positive. (b) Prove that if […] • Solve a Linear Recurrence Relation Using Vector Space Technique Let$V$be a real vector space of all real sequences $(a_i)_{i=1}^{\infty}=(a_1, a_2, \dots).$ Let$U$be a subspace of$V$defined by $U=\{(a_i)_{i=1}^{\infty}\in V \mid a_{n+2}=2a_{n+1}+3a_{n} \text{ for } n=1, 2,\dots \}.$ Let$T$be the linear transformation from […] • Describe the Range of the Matrix Using the Definition of the Range Using the definition of the range of a matrix, describe the range of the matrix $A=\begin{bmatrix} 2 & 4 & 1 & -5 \\ 1 &2 & 1 & -2 \\ 1 & 2 & 0 & -3 \end{bmatrix}.$ Solution. By definition, the range$\calR(A)$of the matrix$A$is given […] • 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 […] • Determine 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 […]