Proving associativity of matrix multiplication

linear-algebra proof-verification

Your proof is fine.

We can change the order of summation as the sum is finite. When we mention multiplication is associative, we might want to mention multiplicative of which object, such as multiplicative of real numbers or complex number.

Related

Motivation/intuition behind using linear algebra behind these combinatorics problem
Is there are similar conjecture like this??
$\dfrac{\partial^2 f}{\partial x \partial y} = 0 \nRightarrow f(x,y) = g(x) + h(y)$
Does the integral cohomology ring determine the ring structure with other coefficients?
Pythagorean triples where the sum of the two cubes is also a square
Coloring $n$ chain with $k$ colors
Can two queens win against a knight on an infinite chess-board?
Packing of parallelograms with sides $1$ and $\sqrt{2}$ and angle $45^{\circ}$ in a rectangular container
Limit of the quotient of sums of recurrent relations
Alice, Bob and his 1956-Triumph [closed]
Sum of determinants of block submatrices
Asymptotic behavior of piecewise recursive random variable.