Simple question related to an ANN diagram showing all of the matrices' dimensions: inputs, weights and output
Yes. reasoning presented here is correct. Mind that $x$ input vector is a row vector (not a column one): $[x_1, x_2, x_3]$. Multiplying $x\cdot w^1$ provides you an another row vector $y_1 = [y_{11}, y_{12}, y_{13}, y_{14}]$ which represents output of neurons 1 to 4 in 1st hidden layer.
Then you multiply $y_1\cdot w^2$ and gain another row vector and so on until the outlayer. At last performing an multiplying 1x4 $\cdot$ 4x1 provides you a scalar which will be a net output after using output's neuron's transfer function on that scalar.