How to move an object's world space matrix to another matrix by linear interpolation?

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I have got the world transform of an object and I want to animate it. I thought about creating a handful of matrices at different positions with different rotations, just like key frames in blender. Now I want to move my Object using its world space to some other matrix' position over time, without just multiplying it with a transform matrix. From there it is supposed to continue to the next one (or "key frame").

From what I found out, the solution would be to linearly interpolate, between the two matrices (or its components), but I just can't figure it out.

What do I need to do, to achieve this simple animation?