How to put 2x2 matrices in the denominator and numerator using MathJax?

Question

Answer(s) to Reciprocal vectors of a two dimensional lattice without borrowing from the third dimension suggest a way forward in figuring out how to divide 2x2 matrices in a straightforward way.

In order to ask a further question on that I'd like to construct a faction like this in MathJax.

and I see the notation used for example in Table 1 of Multi-oriented moiré superstructures of graphene on Ir(111): experimental observations and theoretical models (viewable also: 1, 2) a screen shot is shown below.

Question: If I wanted to generate [[A, B], [C, D]] / [[E, F], [G, H]] as a MathJax fraction with a 2x2 matrix on the top and bottom like these, how would I do it?

note: I don't want to hack the augmented matrix feature, it would be nice if I could use these as real MathJax \frac{}{}tions and perhaps add additional terms.

I also understand that math allows work-arounds where I wouldn't have to write it like this, for example

$$\frac{\mathbf{M_1}+1}{\mathbf{M_2}}$$

but here I'm asking how to do it the hard way, thanks!

  | A  B |        
  |      |                 | A  B |  
  | C  D |                 |      |  / 
 ----------     - OR -     | C  D | /
  | E  F |                         / | E  F |
  |      |                        /  |      |
  | G  H |                           | G  H |

and ideally:

  | A  B |        
  |      | + 1                 | A  B |  
  | C  D |                     |      | + 1 / 
 --------------     - OR -     | C  D |    /
    | E  F |                              / | E  F |
    |      |                             /  |      |
    | G  H |                                | G  H |

Answer 1

\frac{\left[\begin{array}{cc}A&B\\C&D\end{array}\right]+1}{\left[\begin{array}{cc}E&F\\G&H\end{array}\right]} $$\frac{\left[\begin{array}{cc}A&B\\C&D\end{array}\right]+1}{\left[\begin{array}{cc}E&F\\G&H\end{array}\right]}$$

Anything you can create in MathJax (as far as I know) you can put into a fraction using

\frac{Top}{Bottom}.

Answer 2

Using the pmatrix environment (or bmatrix for square braces, but I find the round braces more aesthetically pleasing), the expression with a vinculum may be typeset with the code

$$
\frac{%
\begin{matrix}%
    A & B \\%
    C & D%
\end{pmatrix} + 1%
}{%
\begin{pmatrix}
    D & E \\
    F & G
\end{pmatrix}%
}.
$$

This is rendered as

$$ \frac{% \begin{pmatrix}% A & B \\% C & D% \end{pmatrix} + 1% }{% \begin{pmatrix} E & F \\ G & H \end{pmatrix}% }. $$

Alternatively, the backslash is understood as a delimiter, which can be made large using the \left, \middle, and \right commands. In the style of the screenshot at the end of the question, I might use the code

$$
\left.%
\left[ \begin{pmatrix}%
    A & B \\%
    C & D%
\end{pmatrix} + 1 \right]%
\middle/%
\begin{pmatrix}%
    E & F \\%
    G & H%
\end{pmatrix}
\right..
$$

This is rendered as

$$ \left.% \left[ \begin{pmatrix}% A & B \\% C & D% \end{pmatrix} + 1 \right]% \middle/% \begin{pmatrix}% E & F \\% G & H% \end{pmatrix} \right.. $$

Finally, if you want to make things a little smaller and adjust the vertical alignment a bit, I suppose that you could make the numerator a superscript and the denominator a subscript. To do this, use the code

$$
\left.%
^{%
    \left[ \begin{pmatrix}%
        A & B \\%
        C & D%
    \end{pmatrix} + 1 \right]%
}%
\!\!\middle/\!\!%
_{%
    \begin{pmatrix}%
        E & F \\%
        G & H%
    \end{pmatrix}.%
}%
\right.
$$

Personally, I think that this is rather ugly, but it renders as

$$ \left.% ^{% \left[ \begin{pmatrix}% A & B \\% C & D% \end{pmatrix} + 1 \right]% }% \!\!\middle/\!\!% _{% \begin{pmatrix}% E & F \\% G & H% \end{pmatrix}.% }% \right. $$

All of that having been said, one presumes that $1/M$ is meant to denote the inverse of $M$, presuming that $M$ is invertible. My preference would be to write

$$ \begin{pmatrix} A&B\\C&D \end{pmatrix}\begin{pmatrix} E&F\\G&H\end{pmatrix}^{-1}.$$

Answer 3

As in Xander's answer, / can be used as a delimiter like ()[]|, by which I mean its adjusted by the commands \left, \right, \middle, and also \big, \Big, \bigg, and \Bigg.

For instance, $A/B\big/C\Big/D\bigg/E\Bigg/$ gives $A/B\big/C\Big/D\bigg/E\Bigg/$.

PS the slash slanting the other way can be achieved with $\Bigg\backslash$$\Bigg\backslash$.

Another way to create a slanted line is with the cancel package. Unfortunately it needs something to cancel; so I've used \phantom to make an invisible $2\times 3$ matrix to cancel, then the sub/superscript idea of Xander and some negative \hspace to achieve: $$\require{cancel}M= {}^{\displaystyle \begin{pmatrix}% A & B \\% C & D% \end{pmatrix} + 1 % }% \hspace{-3.5em}\cancel{\phantom{\begin{pmatrix}% A & B & C \\% C & D & C% \end{pmatrix}}}% _{% \displaystyle \hspace{-3em} \begin{pmatrix}% E & F \\% G & H% \end{pmatrix}.% }% $$ Definitely hacky but it seems fine in the three MathJax renderers I tried (SVG, common HTML, and HTML-CSS). Code below-

$$
\require{cancel}
M=
{}^{\displaystyle
     \begin{pmatrix}%
        A & B \\%
        C & D%
    \end{pmatrix} + 1 %
}%
\hspace{-3.5em}\cancel{\phantom{\begin{pmatrix}%
        . & . & . \\%
        . & . & .%
    \end{pmatrix}}}%
_{%
\displaystyle
\hspace{-3em}
    \begin{pmatrix}%
        E & F \\%
        G & H%
    \end{pmatrix}.%
}%
$$