Question Number 204218 by mnjuly1970 last updated on 08/Feb/24
Answered by AST last updated on 08/Feb/24
$$\begin{pmatrix}{\mathrm{3}}&{\mathrm{2}}\\{\mathrm{4}}&{\mathrm{1}}\end{pmatrix}\begin{pmatrix}{{a}}&{{b}}\\{{c}}&{{d}}\end{pmatrix}\begin{pmatrix}{\:{x}}\\{\:{y}}\end{pmatrix}=\begin{pmatrix}{\mathrm{1}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{1}}\end{pmatrix} \\ $$$$\Rightarrow\begin{pmatrix}{\mathrm{3}}&{\mathrm{2}}\\{\mathrm{4}}&{\mathrm{1}}\end{pmatrix}\begin{pmatrix}{{ax}+{by}}\\{{cx}+{dy}}\end{pmatrix}=\begin{pmatrix}{\left(\mathrm{3}{a}+\mathrm{2}{c}\right){x}+\left(\mathrm{3}{b}+\mathrm{2}{d}\right){y}}\\{\left(\mathrm{4}{a}+{c}\right){x}+\left(\mathrm{4}{b}+{d}\right){y}}\end{pmatrix} \\ $$$$\Rightarrow\begin{pmatrix}{\mathrm{3}{a}+\mathrm{2}{c}}&{\mathrm{3}{b}+\mathrm{2}{d}}\\{\mathrm{4}{a}+{c}}&{\mathrm{4}{b}+{d}}\end{pmatrix}=\begin{pmatrix}{\mathrm{1}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{1}}\end{pmatrix}\Rightarrow\begin{pmatrix}{{a}}&{{b}}\\{{c}}&{{d}}\end{pmatrix}=\begin{pmatrix}{\frac{−\mathrm{1}}{\mathrm{5}}}&{\frac{\mathrm{2}}{\mathrm{5}}}\\{\frac{\mathrm{4}}{\mathrm{5}}}&{\frac{−\mathrm{3}}{\mathrm{5}}}\end{pmatrix} \\ $$$$\Rightarrow{A}^{−\mathrm{1}} =\mathrm{5}^{−\mathrm{1}} \begin{pmatrix}{−\mathrm{1}}&{\mathrm{2}}\\{\mathrm{4}}&{−\mathrm{3}}\end{pmatrix}=\begin{pmatrix}{−\mathrm{3}}&{\mathrm{6}}\\{\mathrm{12}}&{−\mathrm{9}}\end{pmatrix}=\begin{pmatrix}{\mathrm{4}}&{\mathrm{6}}\\{\mathrm{5}}&{\mathrm{5}}\end{pmatrix} \\ $$
Answered by Rasheed.Sindhi last updated on 08/Feb/24
$${let}\:{A}^{−\mathrm{1}} =\begin{pmatrix}{{w}}&{{x}}\\{{y}}&{{z}}\end{pmatrix} \\ $$$$\begin{pmatrix}{\mathrm{3}}&{\mathrm{2}}\\{\mathrm{4}}&{\mathrm{1}}\end{pmatrix}\begin{pmatrix}{{w}}&{{x}}\\{{y}}&{{z}}\end{pmatrix}\:\equiv\begin{pmatrix}{\mathrm{1}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{1}}\end{pmatrix}\:\left({mod}\:\mathrm{7}\right) \\ $$$$\begin{pmatrix}{\mathrm{3}{w}+\mathrm{2}{y}}&{\mathrm{3}{x}+\mathrm{2}{z}}\\{\mathrm{4}{w}+{y}}&{\mathrm{4}{x}+{z}}\end{pmatrix}\equiv\begin{pmatrix}{\mathrm{1}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{1}}\end{pmatrix}\:\left({mod}\:\mathrm{7}\right) \\ $$$$\mathrm{Mod}\:\mathrm{7}: \\ $$$$\begin{cases}{\mathrm{3}{w}+\mathrm{2}{y}\equiv\mathrm{1}}\\{\mathrm{4}{w}+{y}\equiv\mathrm{0}}\end{cases}\:\&\:\begin{cases}{\mathrm{3}{x}+\mathrm{2}{z}\equiv\mathrm{0}}\\{\mathrm{4}{x}+{z}\equiv\mathrm{1}}\end{cases}\: \\ $$$$\begin{cases}{\mathrm{3}{w}+\mathrm{2}{y}\equiv\mathrm{1}}\\{\mathrm{8}{w}+\mathrm{2}{y}\equiv\mathrm{0}}\end{cases}\:\&\:\begin{cases}{\mathrm{3}{x}+\mathrm{2}{z}\equiv\mathrm{0}}\\{\mathrm{8}{x}+\mathrm{2}{z}\equiv\mathrm{2}}\end{cases}\: \\ $$$$\mathrm{5}{w}\equiv−\mathrm{1}\Rightarrow\mathrm{5}{w}\equiv\mathrm{6}\Rightarrow{w}\equiv\mathrm{4} \\ $$$${y}\equiv−\mathrm{4}{w}\equiv−\mathrm{4}\left(\mathrm{4}\right)\equiv−\mathrm{16}\equiv\mathrm{5} \\ $$$$\& \\ $$$$\mathrm{5}{x}\equiv\mathrm{2}\Rightarrow{x}\equiv\mathrm{6} \\ $$$${z}\equiv\mathrm{1}−\mathrm{4}{x}=\mathrm{1}−\mathrm{4}\left(\mathrm{6}\right)=−\mathrm{23}\equiv\mathrm{5} \\ $$$${A}^{−\mathrm{1}} \equiv\begin{pmatrix}{{w}}&{{x}}\\{{y}}&{{z}}\end{pmatrix}\:\equiv\begin{pmatrix}{\mathrm{4}}&{\mathrm{6}}\\{\mathrm{5}}&{\mathrm{5}}\end{pmatrix} \\ $$$${Verification}:\: \\ $$$$\begin{pmatrix}{\mathrm{3}}&{\mathrm{2}}\\{\mathrm{4}}&{\mathrm{1}}\end{pmatrix}\begin{pmatrix}{\mathrm{4}}&{\mathrm{6}}\\{\mathrm{5}}&{\mathrm{5}}\end{pmatrix}\:=\begin{pmatrix}{\mathrm{12}+\mathrm{10}}&{\mathrm{18}+\mathrm{10}}\\{\mathrm{16}+\mathrm{5}}&{\mathrm{24}+\mathrm{5}}\end{pmatrix} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\begin{pmatrix}{\mathrm{22}}&{\mathrm{28}}\\{\mathrm{21}}&{\mathrm{29}}\end{pmatrix}\equiv\begin{pmatrix}{\mathrm{1}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{1}}\end{pmatrix} \\ $$