how would i find a matrix with inverse |1 4|
|2 5|
Matrix Inverse -- from Wolfram MathWorld
The 2x2 matrix has an explicit formula for calculating its inverse. Put a,b,c and d back in their original location and then factor out a $\displaystyle \frac{1}{ad-bc}$ from every term.
Jameson gave you references for finding the inverse of a matrix. You are asking "given the inverse, how do I find the matrix" or "Given $\displaystyle A^{-1}$ how do I find A?".
To do that, you need to know that the inverse is "dual"- doing it again puts you back where you started: $\displaystyle A= (A^{-1})^{-1}$.
In other words, just find the inverse of $\displaystyle \begin{bmatrix}1 & 4 \\ 2 & 5\end{bmatrix}$ using the methods Jameson is referring to.
Since this is a simple 2 by 2 problem, you could also do it, right from the definition, like this:
Write the original matrix as $\displaystyle A= \begin{bmatrix}a & b \\ c & d\end{bmatrix}$. The definition of "inverse matrix" says that $\displaystyle AA^{-1}= I$ so we must have
$\displaystyle AA^{-1}= \begin{bmatrix}a & b \\ c & d\end{bmatrix}\begin{bmatrix}1 & 4 \\ 2 & 5\end{bmatrix}= \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$
so
$\displaystyle \begin{bmatrix}a+ 2b & 4a+ 5b \\ c+ 2d & 4c+ 5d\end{bmatrix}= \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$.
That gives you the four equations a+ 2b= 1, 4a+ 5b= 0, c+ 2d= 0, and 4c+ 5d= 1, to solve for a, b, c, and d.