Transition probability matrix

Shares after one week
\(\left( \cdot 6\cdot 4 \right) \left( \begin{matrix} \cdot 7 & \cdot 3 \\ \cdot 6 & \cdot 8 \end{matrix} \right) \)
= (-6\(\times\)·7+-4x·2 ·6\(\times\)·3+·4\(\times\)·8)
= z:(-42+·08 ·18+·32)= (·50 ·50)
\(\Rightarrow\) A = 50% and B = 50%
Shares after two weeks \(\left( \cdot 5\cdot 5 \right) \left( \begin{matrix} \cdot 7 & \cdot 3 \\ \cdot 2 & \cdot 8 \end{matrix} \right) \)
= (-5\(\times\)·7+·5\(\times\).2 ·5\(\times\)·3 +·5\(\times\)·8)
= (-35+·10 ·15 + 40) = (-45 ·55)
A = 45% and B = 55%

Given non-homogeneous equations are
2x + 3y = 5  6X + 5y = 11
\(\Delta =\left| \begin{matrix} 2 & 3 \\ 6 & 5 \end{matrix} \right| =10-18=-8\)
Since \(\Delta \neq 0\) Cramer's rule can be applied and the system is consistent with unique solution
\(\Delta x=\left| \begin{matrix} 5 & 3 \\ 11 & 5 \end{matrix} \right| =25-33=-8\)
\(\Delta y=\left| \begin{matrix} 2 & 5 \\ 6 & 11 \end{matrix} \right| =22-30=-8\)
\(\therefore x=\cfrac { \Delta x }{ \Delta } =\cfrac { -8 }{ -8 } =1\)
\(y=\cfrac { \Delta y }{ \Delta } =\cfrac { -8 }{ -8 } =1\)
\(\therefore \) Solution set is {1, 1}

GivenAX B
\(\left( \begin{matrix} 2 & 4 \\ 4 & 3 \end{matrix} \right) ,\left( \begin{matrix} n \\ 1 \end{matrix} \right) \left( \begin{matrix} 8 \\ 11 \end{matrix} \right) \)
\(\Rightarrow \left( \begin{matrix} 2n+4 \\ 4n+3 \end{matrix} \right) =\left( \begin{matrix} 8 \\ 11 \end{matrix} \right) \)
Equating the corresponding entries on both sides, we get
2n +4 = 8
2n = 8-4
2n=4
\(n=\cfrac { 4 }{ 2 } \)
2 = 2

\(\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} \right] \left[ \begin{matrix} x \\ y \\ z \end{matrix} \right] =\left[ \begin{matrix} 2 \\ -1 \\ 3 \end{matrix} \right] \)
\(\Rightarrow \left( \begin{matrix} x0+0 \\ 0+0+z \\ 0+y+0 \end{matrix} \right) =\left( \begin{matrix} 2 \\ -1 \\ 3 \end{matrix} \right) \)
\(\Rightarrow \left( \begin{matrix} x \\ y \\ z \end{matrix} \right) =\left( \begin{matrix} 2 \\ -1 \\ 3 \end{matrix} \right) \)
\(\Rightarrow x=2\quad z=-1\quad y=3\)
\(\therefore\) Solution set is {2,3, -1}

The non-homogeneous equations are
2x - 3y - 1 = 0, 5x + 2y - 12 = 0
\(\Delta =\left| \begin{matrix} 2 & -3 \\ 5 & 2 \end{matrix} \right| =4+15=19\neq 0\)
Since \(\Delta \neq 0\) Cramer's rule can be applied and the system is consistent with unique solution.
\(\Delta x=\left| \begin{matrix} 1 & -3 \\ 12 & 2 \end{matrix} \right| =2+36=38\)
\(\Delta y=\left| \begin{matrix} 2 & 1 \\ 5 & 12 \end{matrix} \right| =24-5=19\)
\(x=\cfrac { \Delta x }{ \Delta } =\cfrac { 38 }{ 19 } =1\)
\(y=\cfrac { \Delta y }{ \Delta } =\cfrac { 19 }{ 19 } =1\)
\(\therefore \) Solution set is {2, 1}