Can I render a scene "without a camera" by applying a model-view-projection directly to the scene?

Development Rendering
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I am trying to use a custom matrix from a Camera Calibration output. The matrix looks like this:

-4.7415   -2.3964   -2.4075  253.0431
-2.7923    6.1067    6.1569  143.6573
 0.0000    0.0000    1.0000    1.0000
 0.0061   -0.0094    0.0145    1.0000

What would be the proper way to apply this to the whole scene (as a camera would) in order to render objects in a certain way?

This is a reference from a sketch I have made in processing, showing the result I would like to achieve.