cmi-entrance

Papers (31)
2025
mscds 11
2024
mscds 10 pgmath 1 ugmath 20
2023
pgmath 3 ugmath 16
2022
pgmath 2 ugmath_22may 16 ugmath_23may 16
2021
pgmath 1 ugmath 15
2020
ugmath 15
2019
pgmath 2 ugmath 10
2018
pgmath 2 ugmath 14
2017
pgmath 1 ugmath 13
2016
pgmath 2 ugmath 11
2015
pgmath 1 ugmath 13
2014
ugmath 9
2013
pgmath 8 ugmath 10
2012
pgmath 2 ugmath 14
2011
pgmath 1 ugmath 12
2010
pgmath 4 ugmath 19
2016 pgmath

2 maths questions

Q18* 10 marks Groups Group Actions and Surjectivity/Injectivity of Maps View
Let $G$ be a subgroup of the group of permutations on a finite set $X$. Let $F$ be the $\mathbb{C}$-vector-space of all the functions from $X$ to $\mathbb{C}$. $G$ acts on $F$ by $(g \cdot f) : x \mapsto f(g^{-1}(x))$. Show that there is a $\phi \in F$ such that $g \cdot \phi = \phi$ for every $g \in G$. Show that there is a subspace $F'$ of $F$ such that $F = F' \oplus \mathbb{C}\langle \phi \rangle$ and such that $g \cdot f \in F'$ for every $g \in G$ and $f \in F'$.
Q19* 10 marks Matrices Structured Matrix Characterization View
(A) Let $A$ and $B$ be $n \times n$ matrices with entries in $\mathbb{N}$. Show that if $B = A^{-1}$ then $A$ and $B$ are permutation matrices. (A permutation matrix is a matrix obtained by permuting the rows of the identity matrix.)
(B) Let $A$ be an $n \times n$ complex matrix that is not a scalar multiple of $I_n$. Show that $A$ is similar to a matrix $B$ such that $B_{1,1}$ (i.e. the top left entry of $B$) is 0.