inverse matrices

inverse matrices

(b)

Hence find the inverse, $A_n^{-1}$ of $A_n$ in terms of $n$

Formula inverse matrice:

$\begin{array}{|rcll|} \hline A &=& e\cdot \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} \\\\ A^{-1} &=& \dfrac{e\cdot \begin{pmatrix} d & -b \\ -c & a \\ \end{pmatrix}}{ det(A)} \qquad det(A) = e^2\cdot(a\cdot d - c\cdot b) \\\\ && \mathbf{ \boxed{ A^{-1} = \dfrac{ \begin{pmatrix} d & -b \\ -c & a \\ \end{pmatrix}}{e\cdot(a\cdot d - c\cdot b)} } } \\ \hline \end{array}$

$\begin{array}{|rcll|} \hline A_n &=& 5^{n-1} \begin{pmatrix} 2n+5 & -2n \\ 2n & 5-2n \\ \end{pmatrix} \\\\ A_n^{-1} &=& \dfrac{ \begin{pmatrix} 5-2n & 2n \\ -2n & 2n+5 \\ \end{pmatrix} } {5^{n-1}\cdot \Big((2n+5)(5-2n) - (2n)(-2n) \Big) } \\\\ &=& \dfrac{ \begin{pmatrix} 5-2n & 2n \\ -2n & 2n+5 \\ \end{pmatrix} } {5^{n-1}\cdot (25-4n^2+4n^2 ) } \\\\ &=& \dfrac{ \begin{pmatrix} 5-2n & 2n \\ -2n & 2n+5 \\ \end{pmatrix} } { 5^{n-1}\cdot 5^2 } \\\\ &=& \dfrac{ \begin{pmatrix} 5-2n & 2n \\ -2n & 2n+5 \\ \end{pmatrix} } { 5^{n-1+2} } \\\\ &=& \dfrac{ \begin{pmatrix} 5-2n & 2n \\ -2n & 2n+5 \\ \end{pmatrix} } { 5^{n+1} } \\\\ \mathbf{A_n^{-1}} & \mathbf{=} & \mathbf{ 5^{-n-1} \begin{pmatrix} 5-2n & 2n \\ -2n & 2n+5 \\ \end{pmatrix} } \\ \hline \end{array}$