There are three types matrices that we are interested in: the transfer matrix, the refraction matrix, and the system matrix.
The transfer matrix involves, as its name suggests, movement from one place to another. We use this matrix to transfer the light from the object to the first lens, between each lens, and from the last lens to the image. (At this time I would like to inform you that I have chosen to limit my model to thin lenses only. The reason for this is, with thick lenses we would need a transfer matrix within each lens. In our system of three lenses, that means we would have to include three more matrices that we don't want to deal with.)
The refraction matrix gives us the information regarding how much the light will bend when it hits the surface of the lens. Each lens will have its own refraction matrix. Whereas the important term with the transfer matrix is a distance, the key piece of information in the refraction matrix is the focal length of the lens.
The system matrix is the product of all its parts. To create a system matrix for our three-lens system, we multiply together (from back to front) the transfer and refraction matrices. Since we are combining seven matrices, the system matrix becomes a pretty ugly thing. It is a good thing that we only need to focus on two entries in the system matrix - the top right and the bottom left.
System Matrix - Bottom Left
System Matrix - Top Right
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