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.