# Thread: finding a matrix given the inverse

1. ## finding a matrix given the inverse

how would i find a matrix with inverse |1 4|
|2 5|

2. Originally Posted by cottekr
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 $\frac{1}{ad-bc}$ from every term.

3. 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 $A^{-1}$ 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: $A= (A^{-1})^{-1}$.

In other words, just find the inverse of $\begin{bmatrix}1 & 4 \\ 2 & 5\end{bmatrix}$ 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 $A= \begin{bmatrix}a & b \\ c & d\end{bmatrix}$. The definition of "inverse matrix" says that $AA^{-1}= I$ so we must have

$AA^{-1}= \begin{bmatrix}a & b \\ c & d\end{bmatrix}\begin{bmatrix}1 & 4 \\ 2 & 5\end{bmatrix}= \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$

so
$\begin{bmatrix}a+ 2b & 4a+ 5b \\ c+ 2d & 4c+ 5d\end{bmatrix}= \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$.

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.