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Inverse Matrices Problems and Solutions


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  • A Rational Root of a Monic Polynomial with Integer Coefficients is an IntegerA 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 DiagonalizableA 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?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 NormalA 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 1Eigenvalues 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 2018Mathematics 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 ZeroA 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 GroupsSurjective 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 […]

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