# inverse-matrix

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• A Rational Root of a Monic Polynomial with Integer Coefficients is an Integer Suppose that $\alpha$ is a rational root of a monic polynomial $f(x)$ in $\Z[x]$. Prove that $\alpha$ is an integer.   Proof. Suppose that $\alpha=\frac{p}{q}$ is a rational number in lowest terms, that is, $p$ and $q$ are relatively prime […]
• A Matrix Similar to a Diagonalizable Matrix is Also Diagonalizable Let $A, B$ be matrices. Show that if $A$ is diagonalizable and if $B$ is similar to $A$, then $B$ is diagonalizable.   Definitions/Hint. Recall the relevant definitions. Two matrices $A$ and $B$ are similar if there exists a nonsingular (invertible) matrix $S$ such […]
• Is the Derivative Linear Transformation Diagonalizable? Let $\mathrm{P}_2$ denote the vector space of polynomials of degree $2$ or less, and let $T : \mathrm{P}_2 \rightarrow \mathrm{P}_2$ be the derivative linear transformation, defined by $T( ax^2 + bx + c ) = 2ax + b .$ Is $T$ diagonalizable? If so, find a diagonal matrix which […]
• A Subgroup of the Smallest Prime Divisor Index of a Group is Normal Let $G$ be a finite group of order $n$ and suppose that $p$ is the smallest prime number dividing $n$. Then prove that any subgroup of index $p$ is a normal subgroup of $G$.   Hint. Consider the action of the group $G$ on the left cosets $G/H$ by left […]
• Eigenvalues of a Stochastic Matrix is Always Less than or Equal to 1 Let $A=(a_{ij})$ be an $n \times n$ matrix. We say that $A=(a_{ij})$ is a right stochastic matrix if each entry $a_{ij}$ is nonnegative and the sum of the entries of each row is $1$. That is, we have $a_{ij}\geq 0 \quad \text{ and } \quad a_{i1}+a_{i2}+\cdots+a_{in}=1$ for $1 […] • Mathematics About the Number 2018 Happy New Year 2018!! Here are several mathematical facts about the number 2018. Is 2018 a Prime Number? The number 2018 is an even number, so in particular 2018 is not a prime number. The prime factorization of 2018 is $2018=2\cdot 1009.$ Here$2$and$1009$are […] • A Linear Transformation is Injective (One-To-One) if and only if the Nullity is Zero Let$U$and$V$be vector spaces over a scalar field$\F$. Let$T: U \to V$be a linear transformation. Prove that$T$is injective (one-to-one) if and only if the nullity of$T$is zero. Definition (Injective, One-to-One Linear Transformation). A linear […] • Surjective Group Homomorphism to$\Z$and Direct Product of Abelian Groups Let$G$be an abelian group and let$f: G\to \Z$be a surjective group homomorphism. Prove that we have an isomorphism of groups: $G \cong \ker(f)\times \Z.$ Proof. Since$f:G\to \Z$is surjective, there exists an element$a\in G\$ such […]