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Let C=AB. To get the c11 element of C i do a dot product of row1 of A and column 1 of B.

Row 1 contains x, y, z component of a vector of A.

Column 1 contains x components of three vector of B.

C specifies a parallelopiped whose volume is the product of volume of parallelopiped specified by A, B.

Since det A*det B=det AB = det C.

C11 is a x component of one of the vector of the C.

How come the dot product of x, y, z components of a vector of A and the x components of three vector of B gives the x component of a vector of C?

I know that matrix represent a linear system and when you multiply two matrice, it means your taking composition of two linear function so the matrix mutilipication is in that way.

But how can we prove in terms of vector? Since vetors are imagnabile, the proof of matrix mutilipication interms of vector can gives us more intuition regarding the matrix mutilipication.