# Transition Matrix Calculator

The calculator will find the transition matrix from the first basis to the second basis, with steps shown.

## Your Input

**Calculate the transition matrix from $$$\left[\begin{array}{cc}-3 & 4\\2 & -2\end{array}\right]$$$ to $$$\left[\begin{array}{cc}-1 & 2\\2 & -2\end{array}\right]$$$.**

## Solution

To find the transition matrix, augment the matrix of the second basis with the matrix of the first basis and perform row operations trying to make the identity matrix to the left. Then to the right will be the transition matrix.

So, augment the matrix of the second basis with the matrix of the first basis:

$$$\left[\begin{array}{cc|cc}-1 & 2 & -3 & 4\\2 & -2 & 2 & -2\end{array}\right]$$$

Multiply row $$$1$$$ by $$$-1$$$: $$$R_{1} = - R_{1}$$$.

$$$\left[\begin{array}{cc|cc}1 & -2 & 3 & -4\\2 & -2 & 2 & -2\end{array}\right]$$$

Subtract row $$$1$$$ multiplied by $$$2$$$ from row $$$2$$$: $$$R_{2} = R_{2} - 2 R_{1}$$$.

$$$\left[\begin{array}{cc|cc}1 & -2 & 3 & -4\\0 & 2 & -4 & 6\end{array}\right]$$$

Divide row $$$2$$$ by $$$2$$$: $$$R_{2} = \frac{R_{2}}{2}$$$.

$$$\left[\begin{array}{cc|cc}1 & -2 & 3 & -4\\0 & 1 & -2 & 3\end{array}\right]$$$

Add row $$$2$$$ multiplied by $$$2$$$ to row $$$1$$$: $$$R_{1} = R_{1} + 2 R_{2}$$$.

$$$\left[\begin{array}{cc|cc}1 & 0 & -1 & 2\\0 & 1 & -2 & 3\end{array}\right]$$$

We are done. On the left is the identity matrix. On the right is the transition matrix.