MatrixSolutions

The general idea of a matrix inverse is that if A is a matrix, its inverse, $\mathbf {X} ^{-1}$ 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

$\mathbf {X} =\left[{\begin{array}{cc}1&4\\-1&2\end{array}}\right]^{-1}$

I think that the easiest solution is to solve as a set of simultaneous equations. If $\mathbf {X} =\left[{\begin{array}{cc}a&b\\c&d\end{array}}\right]$,

$\left[{\begin{array}{cc}1&4\\-1&2\end{array}}\right]\left[{\begin{array}{cc}a&b\\c&d\end{array}}\right]=\left[{\begin{array}{cc}1&0\\0&1\end{array}}\right]$

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.