Low confidence that this will help, but my approach: I mentally move the right-hand matrix up, so that the space "in between" them (right of the first, below the second) is the right shape for the result. Each value of the result is the dot product of the vectors to the left and above it. (I don't have a trick for dot products, I just know how to calculate them.)

. . . . g h i
a b c * j k l
d e f . m n o

"becomes"

. . . g h i
. . . j k l
. . . m n o
. . . -----
a b c|S T W
d e f|X Y Z

and e.g. S is (a b c) dot (g j m), Y is (d e f) dot (h k n).