# system-of-linear-equations-eye-catch

• Number Theoretical Problem Proved by Group Theory. $a^{2^n}+b^{2^n}\equiv 0 \pmod{p}$ Implies $2^{n+1}|p-1$. Let $a, b$ be relatively prime integers and let $p$ be a prime number. Suppose that we have $a^{2^n}+b^{2^n}\equiv 0 \pmod{p}$ for some positive integer $n$. Then prove that $2^{n+1}$ divides $p-1$.   Proof. Since $a$ and $b$ are relatively prime, at least one […]
• Overall Fraction of Defective Smartphones of Three Factories A certain model of smartphone is manufactured by three factories A, B, and C. Factories A, B, and C produce $60\%$, $25\%$, and $15\%$ of the smartphones, respectively. Suppose that their defective rates are $5\%$, $2\%$, and $7\%$, respectively. Determine the overall fraction of […]
• The Transpose of a Nonsingular Matrix is Nonsingular Let $A$ be an $n\times n$ nonsingular matrix. Prove that the transpose matrix $A^{\trans}$ is also nonsingular.   Definition (Nonsingular Matrix). By definition, $A^{\trans}$ is a nonsingular matrix if the only solution to […]
• Given the Data of Eigenvalues, Determine if the Matrix is Invertible In each of the following cases, can we conclude that $A$ is invertible? If so, find an expression for $A^{-1}$ as a linear combination of positive powers of $A$. If $A$ is not invertible, explain why not. (a) The matrix $A$ is a $3 \times 3$ matrix with eigenvalues $\lambda=i , […] • A Group Homomorphism and an Abelian Group Let$G$be a group. Define a map$f:G \to G$by sending each element$g \in G$to its inverse$g^{-1} \in G$. Show that$G$is an abelian group if and only if the map$f: G\to G$is a group homomorphism. Proof.$(\implies)$If$G$is an abelian group, then$f$[…] • Powers of a Diagonal Matrix Let$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 […]
• The Additive Group of Rational Numbers and The Multiplicative Group of Positive Rational Numbers are Not Isomorphic Let $(\Q, +)$ be the additive group of rational numbers and let $(\Q_{ > 0}, \times)$ be the multiplicative group of positive rational numbers. Prove that $(\Q, +)$ and $(\Q_{ > 0}, \times)$ are not isomorphic as groups.   Proof. Suppose, towards a […]
• 5 is Prime But 7 is Not Prime in the Ring $\Z[\sqrt{2}]$ In the ring $\Z[\sqrt{2}]=\{a+\sqrt{2}b \mid a, b \in \Z\},$ show that $5$ is a prime element but $7$ is not a prime element.   Hint. An element $p$ in a ring $R$ is prime if $p$ is non zero, non unit element and whenever $p$ divide $ab$ for $a, b \in R$, then $p$ […]