You did not provide 't20' or a description of its structure, so I had to make my own.

Hopefully it is compatible with yours:

t20 = Table[

"inert"[{x, y, Sin[x Sin[y]]}], {x, 1, 10, .1}, {y, 1, 10, .1}];

t20 = Cases[t20, "inert"[a___] :> a, Infinity];

I think I achieved what you wanted with the following:

Manipulate[

If[XorY == "y=c",

Grid[{{ListLinePlot[Cases[t20, {x_, c, z_} :> {x, z}],

AxesLabel -> {"x", "z"}, PlotStyle -> Thick,

ImageSize -> {275, 275}],

ListPlot3D[t20, Mesh -> {{0}, {{c, Thick}}},

ImageSize -> {275, 275}]}}],

Grid[{{ListLinePlot[Cases[t20, {c, y_, z_} :> {y, z}],

AxesLabel -> {"y", "z"}, PlotStyle -> Thick,

ImageSize -> {275, 275}],

ListPlot3D[t20, Mesh -> {{{c, Thick}}, {0}},

ImageSize -> {275, 275}]}}]], {{XorY, "x=c",

"direction"}, {"x=c", "y=c"},

ControlType -> Setter}, {{c, 1., "c"}, 1., 10., .1,

Appearance -> "Labeled"}]

A matrix uses the Part[] or [[]] syntax. However you do not seem to have a matrix. You seem to have a list constaining lists of three coordinates.

That is, instead of having something such as {{1,2,3},{4,5,6}}, you have the equivalent information in a different form: {{1,1,1},{1,2,2},{1,3,3},{2,1,4},{2,2,5},{2,3,6}}.

Cases[] would be a way to extract just the coordinates that you are interested in, such as the ones where the first coordinate (the 'x' coordinate) is equal to a constant 'c'.

I would try to present this better but I am out of time for now. I hope it helps you.