It's not the same, but in most cases a matrix can be implemented through a 3D LUT.
Basically, a matrix is used to "cross pollinate" the three color channels. Each channel is allowed to have a mixture of the other two as well as its own component. So the Red channel, for instance, can contain the information from the Red channel as well as a portion of the information from the Blue and Green channels. Each of the three channels, in effect, is given three components. All of this is useful because in terms of how a camera works, it is almost impossible to ensure that each channel is getting pure information. Crosstalk is inherent in the capture process, due to numerous factors including but not limited to impure color dyes in the color filters present on the sensor (in the case of a sensor with a color filter array, such as a Bayer pattern sensor), imperfections in the dichroics of a 3 chip camera design, lack of perfect focus on to the underlying pixels in a color filter array sensor, and imperfections in the optical elements. The use of a matrix allows minimization of crosstalk by analyzing the sensor's output and applying information from all three channels to each of the three channels in proper proportions, sometimes using mathematical expressions for more specific control (that's what Peter is talking about). When done correctly (often by the manufacturer, as they have much better knowledge of the sensor design) this yields purer colors and restores "normal" saturation. Graeme can explain this in more depth, but basically the colorimetry of Red images has improved in large part due to his updating of the color matrix, which in the case of Red is currently presented to you in the form of Redcolor, Redcolor2, and Redcolor3. In the early days of Red's cameras, the saturation was typically much too low and the colors were not as "true to life" as they are today, in large part due to the lack of a well evolved color matrix. By carefully altering the proportions I'm talking about, both color purity and saturation have been altered based on feedback from users, in order to provide more pleasing results.
In a post environment, matrixing can be used exactly as Peter describes, to "normalize" a signal that is "wrong", such as an image that was shot with the wrong white balance. One fairly accessible implementation of this is DaVinci Resolve's Color Mixer, which is essentially a variable matrix. I've used that tool to alter entire scenes that were shot with improper white balance, resulting in a very blue image (the color temperature was incorrectly set at around 3200K, which is roughly equivalent to tungsten lighting, even though the scene was a day exterior). By populating the deficient Red channel with information from the Blue channel, and minimizing the blue component of the blue channel, as well as some other channel manipulations, I was able to "normalize" the image in a way that was not really achievable by changing normal balance. Essentially, I used the matrix as a color temperature/tint control, using the blue channel component to alter temperature, and the green channel to alter tint (essentially a change on the green/magenta axis).
I hope that's not overly confusing. A color matrix can be a very useful tool in the right situation and in the right hands. It is not a "normal" color correction tool (unless you're a video engineer), but in some situations it can definitely do things that other tools are not easily capable of.