I need to find a matrix which would solve for the following:
$$\begin{bmatrix}4 & 7\\4 & 2 \end{bmatrix} * \begin{bmatrix}- & -\\- &- \end{bmatrix} = \begin{bmatrix}1 & 0\\0 & 1 \end{bmatrix}$$
I tried using the inverse which gave me $ \begin{bmatrix}\frac{2}{-20} & \frac{-7}{-20}\\\frac{-4}{-20} & \frac{4}{-20} \end{bmatrix} $
Also attempted making $\Bigg [\begin{matrix}4 & 7\\4 & 2 \end{matrix}\Bigg | \begin{matrix}1 & 0\\0 & 1 \end{matrix}\Bigg ]$
which gave me the following: $\Bigg [\begin{matrix}1 & 0\\0 & 1 \end{matrix}\Bigg | \begin{matrix}\frac17 & 0\\\frac1{20} & \frac1{-5} \end{matrix}\Bigg ]$
I am guessing I did something wrong at some step but Cannot seem to figure out where!
Best Answer
I have a trick for you. Let $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ be any matrix. Then its inverse is $$ \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a\end{bmatrix}$$