Q-3: $$ [\mathbf{+}+\mathbf{+}+\mathbf{5} $$ marks $$ ] $$ a) Consider the linear system $$ \left\{\begin{array}{c}x-y-3 z=3 \ 2 x+z=0 \ 2 y+7 z=c\end{array} ight. $$ i) For what value of $$ c $$ does the system have a solution? ii) Solve the system at the value of $$ c $$ from part (i)? b) Let $$ A=\left[\begin{array}{cc}1 & \frac{1}{3} \ c & d\end{array} ight] $$. Find $$ c $$ and $$ d $$ such that $$ A^{2}=-I_{2} $$. c) If $$ \left|\begin{array}{ccc}1 & t & t^{2} \ t & 1 & t \ t^{2} \ t^{2} & t & 1 \ t^{3} & t^{2} & t \ t^{2}\end{array} ight|=(1-f(t))^{p} $$. Find the function $$ f(t) $$ and the power $$ p $$ ?

Question

Q-3: $$ [\mathbf{+}+\mathbf{+}+\mathbf{5} $$ marks $$ ] $$ a) Consider the linear system $$ \left\{\begin{array}{c}x-y-3 z=3 \ 2 x+z=0 \ 2 y+7 z=c\end{array}ight. $$ i) For what value of $$ c $$ does the system have a solution? ii) Solve the system at the value of $$ c $$ from part (i)? b) Let $$ A=\left[\begin{array}{cc}1 & \frac{1}{3} \ c & d\end{array}ight] $$. Find $$ c $$ and $$ d $$ such that $$ A^{2}=-I_{2} $$. c) If $$ \left|\begin{array}{ccc}1 & t & t^{2} \ t & 1 & t \ t^{2} \ t^{2} & t & 1 \ t^{3} & t^{2} & t \ t^{2}\end{array}ight|=(1-f(t))^{p} $$. Find the function $$ f(t) $$ and the power $$ p $$ ?

Chandler Mejia · Accepted Answer

a) i) $$ c=12 $$. ii) $$ x=4, y=-4, z=2 $$. b) $$ c=-6, d=-4 $$. c) The function $$ f(t) $$ and the power $$ p $$ are not provided in the simplified answer.

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