4x4 matrix with colours and symbols - no repeat in row or column

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I was watching Survivor (reality show) and they got a following puzzle. There is 4x4 matrix where each column has a color and each row has a symbol.

By moving around or replacing you should get to the point, where in each row or column there is no repeating symbol or repeating color.

I was able to solve that, but I cannot explain how. I think this is some variant of Algebraic magic square. But is there some algorithm which I could always follow to get to the final required state?