When we add two matrices we just simply add the corresponding elements but when we multiply two matrices there is a much more complex process.Why does it happens?
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1Duplicate of Why, historically, do we multiply matrices as we do?, Intuition behind Matrix Multiplication, and Fast(est) and intuitive ways to look at matrix multiplication? – complexist May 26 '14 at 07:40
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It's just a definition. We define matrix sum in a useful way (which turns out to be easy) and matrix multiplication in a useful way (which turns out to be a bit more complicated). See above for more. – Gamma Function May 26 '14 at 07:41
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Because we define operations in a way that is useful, you could multiply every entry in a matrix with the corresponding entry in the other matrix (also known as Hadamard product), but the usual definition turns out to be very useful for how it relates to linear equations – Alessandro Codenotti May 26 '14 at 07:42
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An analogy is with polynomials, easier to add than to multiply (or even with addition of decimal numbers vs. multiplying them). A deeper analogy would be adding polynomials (add like terms) vs. composition of polynomials (in which coefficients in the result depend widely on the coefficients of varying degrees in the operands). – hardmath May 26 '14 at 07:46