So, I was studying coupled oscillations and came across a statement that I couldn't figure out. It was that a particular matrix was symmetrical by Newton's Third Law. I know what Newton's Third Law is, I know what symmetric matrix is. But, for example, a matrix like this: -2k/m k/m k/m -2k/m Being multiplied times a vector like <x1,x2> to produce the acceleration vector <x1'',x2''>. It comes from the equations x1''=(-2k/m)x1+(k/m)x2 x2''=(k/m2)x1+(-2k/m)x2 I'm trying to see how Newton's third law makes the matrix symmetrical. I mean, I can see why each mass's equation of acceleration takes the same form, because the choice of x1 being x1 and x2 being x2 is arbitrary. Can somebody explain how the force-pair law means that this matrix will be symmetrical?