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ISI2018-PCB-A1

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akash.dinkar12 asked May 12, 2019
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ISI2018-PCB-A4
Let $A$ and $B$ are two non-empty finite subsets of $\mathbb{Z}$, the set of all integers. Define $A+B=\{a+b:a\in A,b\in B\}$.Prove that $\mid A+B \mid \geq \mid A \mid + \mid B \mid -1 $, where $\mid S \mid$ denotes the cardinality of finite set $S$.
Let $A$ and $B$ are two non-empty finite subsets of $\mathbb{Z}$, the set of all integers. Define $A+B=\{a+b:a\in A,b\in B\}$.Prove that $\mid A+B \mid \geq \mid A \mid ...
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akash.dinkar12 asked May 12, 2019
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ISI2018-PCB-A3
Let $n,r\ $and$\ s$ be positive integers, each greater than $2$.Prove that $n^r-1$ divides $n^s-1$ if and only if $r$ divides $s$.
Let $n,r\ $and$\ s$ be positive integers, each greater than $2$.Prove that $n^r-1$ divides $n^s-1$ if and only if $r$ divides $s$.
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admin asked Aug 8, 2022
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ISI 2020 | PCB Mathematics | Question: 3
Suppose $A$ is an $(n \times n)$ matrix over $\mathbb{R}$ such that $A^{p}=0$ for some positive integer $p$. Prove that $I+A$ is an invertible matrix, where $I$ is the $(n \times n)$ identity matrix. Find the characteristic polynomial of $A$.
Suppose $A$ is an $(n \times n)$ matrix over $\mathbb{R}$ such that $A^{p}=0$ for some positive integer $p$.Prove that $I+A$ is an invertible matrix, where $I$ is the $(n...
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