cmi-entrance 2013 QB2

cmi-entrance · India · pgmath 10 marks Matrices Matrix Group and Subgroup Structure

(a) Show that there exists a $3 \times 3$ invertible matrix $M \neq I _ { 3 }$ with entries in the field $\mathbb { F } _ { 2 }$ such that $M ^ { 7 } = I _ { 3 }$.
(b) Let $A$ be an $m \times n$ matrix, and $\mathbf { b }$ an $m \times 1$ vector, both with integer entries.

Suppose that there exists a prime number $p$ such that the equation $A \mathbf { x } = \mathbf { b }$ seen as an equation over the finite field $\mathbb { F } _ { p }$ has a solution. Then does there exist a solution to $A \mathbf { x } = \mathbf { b }$ over the real numbers?
If $A \mathbf { x } = \mathbf { b }$ has a solution over $\mathbb { F } _ { p }$ for every prime $p$, is a real solution guaranteed?

(a) Show that there exists a $3 \times 3$ invertible matrix $M \neq I _ { 3 }$ with entries in the field $\mathbb { F } _ { 2 }$ such that $M ^ { 7 } = I _ { 3 }$.\\
(b) Let $A$ be an $m \times n$ matrix, and $\mathbf { b }$ an $m \times 1$ vector, both with integer entries.
\begin{enumerate}
\item Suppose that there exists a prime number $p$ such that the equation $A \mathbf { x } = \mathbf { b }$ seen as an equation over the finite field $\mathbb { F } _ { p }$ has a solution. Then does there exist a solution to $A \mathbf { x } = \mathbf { b }$ over the real numbers?
\item If $A \mathbf { x } = \mathbf { b }$ has a solution over $\mathbb { F } _ { p }$ for every prime $p$, is a real solution guaranteed?
\end{enumerate}

Paper Questions

QA1 QA10 QA11 QA12 QA13 QA14 QA15 QA2 QA3 QA4 QA5 QA6 QA7 QA8 QA9 QB1 QB10 QB2 QB3 QB4 QB5 QB6 QB7 QB8 QB9