how would i find a matrix with inverse |1 4|
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 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: .
In other words, just find the inverse of 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 . The definition of "inverse matrix" says that so we must have
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.