Can a dense layer on many inputs be represented as a single matrix multiplication?

Denote a[2, 3] to be a matrix of dimension 2x3. Say there are 10 elements in each input and the network is a two-element classifier (cat or dog, for example). Say there is just one dense layer. For now I am ignoring the bias vector. I know this is an over-simplified neural net, but it is just for this example. Each output in a dense layer of a neural net can be calculated as

output = matmul(input, weights)

Where weights is a weight matrix 10x2, input is an input vector 1x10, and output is an output vector 1x2.

My question is this: Can an entire series of inputs be computed at the same time with a single matrix multiplication? It seems like you could compute

output = matmul(input, weights)

Where there are 100 inputs total, and input is 100x10, weights is 10x2, and output is 100x2.

In back propagation, you could do something similar:

input_err = matmul(output_err, transpose(weights))
weights_err = matmul(transpose(input), output_err)
weights -= learning_rate*weights_err

Where weights is the same, output_err is 100x2, and input is 100x10.

However, I tried to implement a neural network in this way from scratch and I am currently unsuccessful. I am wondering if I have some other error or if my approach is fundamentally wrong.

Thanks!