Question

Find the inverse of matrix [3-1-10 9]

Answer 1

$\mathbf A=\begin{bmatrix}3&-1\\-10&9\end{bmatrix}$

The inverse is given by 

$\mathbf A^{-1}=\dfrac1{\det\mathbf A}\mathrm{adj}\,\mathbf A$

where $\det\mathbf A$ is the determinant of the matrix, and $\mathrm adj\,\mathbf A$ is the adjugate matrix, or transpose of the cofactor matrix.

$\det\mathbf A=\begin{vmatrix}3&-1\\-10&9\end{vmatrix}=3(9)-(-1)(-10)=17$

$\mathrm{adj}\,\mathbf A=\begin{bmatrix}9&10\\1&3\end{bmatrix}^\top=\begin{bmatrix}9&1\\10&3\end{bmatrix}$

So the inverse is

$\mathbf A^{-1}=\dfrac1{17}\begin{bmatrix}9&1\\10&3\end{bmatrix}$