MatrixSolutions
From Brain Mapping
The general idea of a matrix inverse is that if A is a matrix, its inverse, 
is the matrix which, when multiplied by A yields the identity matrix, I. I, in turn, is the matrix which, when multiplied by any matrix B just gives back B.
For the problem posed in the quiz:
find
![\mathbf{X}=\left[\begin{array}{cc} 1 & 4\\ -1 & 2\end{array}\right]^{-1}](/wiki/images/math/7/d/2/7d28546b18b3ee7317ff982d2ba01e56.png)
I think that the easiest solution is to solve as a set of simultaneous equations. If ![\mathbf{X} = \left[\begin{array}{cc} a & b\\ c & d\end{array}\right]](/wiki/images/math/3/b/1/3b1b19520c98940abca2cd8f3b2c5f09.png)
,
![\left[\begin{array}{cc} 1 & 4\\ -1 & 2\end{array}\right] \left[\begin{array}{cc} a & b\\ c & d\end{array}\right] = \left[\begin{array}{cc} 1 & 0\\ 0 & 1\end{array}\right]](/wiki/images/math/8/a/6/8a6adec078f01990662ba47fc2e1ee73.png)
Thus, the rules of matrix multiplication give us this set of four equations:
- a + 4c = 1
- b + 4d = 0
- − a + 2c = 0
- − b + 2d = 1
which is very easy to solve by substitution:
- b = − 4d
- a = 2c
- 2c + 4c = 1
- c = 1 / 6
- etc,...
Of course, you can use Cramer's rule if you remember it, or some other solving algorithm. I think you can get a matrix solution app for your iPhone, as well.
We will use this general concept a lot in Principles of Neuroimaging. In particular, when we imagine the image result of an experiment to be the sum of a variety of influences (experimental and otherwise), we will use matrix form to evaluate the strength of each of these influences in creating our experimental result.
