A MatlabMatrix is a matrix defined in MATLAB® memory space. In Maple, the matrix is always named as a string, with double quotes. For example, Matlab[inv]("M") computes the inverse of the variable, "M", which is defined in MATLAB®. If a variable M is defined in Maple, it will be unaffected by, and have no effect on the stated call.  The two calls, Matlab[inv](M) and Matlab[inv]("M") are very different.

A MapleMatrix is any Maple expression that is equivalent to a matrix. For example Array(1..2,[1,2]) is equivalent to [1,2], or Vector([1,2]). A MapleMatrix can be an rtable (Array, Matrix or Vector), a table, or a constant (numeric or complex numeric or a symbolic constant such as Pi or infinity). All elements of the MapleMatrix must be of type constant.

MATLAB® has access to variables defined in its memory space, and Maple has access to variables defined in its memory space.  Variables are not automatically  shared between memory spaces; they must be explicitly set or read.

For example, if a user starts Maple and defines a matrix "M" by typing M := Matrix([[1,2],[3,4]]); the memory space looks like this.

If that user then opens a link to MATLAB® and sets a variable, "X", in MATLAB® to the same value as "M" (using the command Matlab[setvar]("X", M);), the memory is as follows.

The variable M is a MapleMatrix, and "X" is a MatlabMatrix. The command Matlab[inv](M) copies the matrix M to MATLAB® memory space, defining a temporary variable result_for_maple in the process. MATLAB® computes the inverse of result_for_maple and sends the result back to Maple.

The command Matlab[inv]("X") computes the inverse of "X" and sends the result to Maple.

It is possible to define the name "M" in both Maple and MATLAB®. After the command Matlab[setvar]("M", Pi), the memory spaces would be as follows.

Consider the two commands Matlab[det](M), and Matlab[det]("M"). The subtle difference between these commands makes a significant difference in their results. The first command returns the determinant of the 2x2 matrix, M=[[1,2],[3,4]], which is -2. The second command returns the determinant of the 1x1 matrix, "M"=[Pi], which is Pi.

$M≔\mathrm{Matrix}\left(\left[\left[1\,2\right]\,\left[3\,4\right]\right]\right)$

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

$\mathrm{with}\left(\mathrm{Matlab}\right)\:$

$\mathrm{Matlab}\left[\mathrm{setvar}\right]\left(''X''\,M\right)$

$\mathrm{Matlab}\left[\mathrm{setvar}\right]\left(''M''\,\mathrm{\pi }\right)$

$\mathrm{Matlab}\left[\det \right]\left(M\right)$

$\mathrm{Matlab}\left[\det \right]\left(''M''\right)$