Other than notation devices ...
Matrices are far more than mere notational devices, but even if they were, don't deride notational devices! More compact notation simplifies writing, simplifies reading, simplifies thinking, and because of that, it enables new ways of thinking.
Think of the progress from marks on a stick to representing numbers by devices such as MMXIV to place value notation such as 2014. Even those marks on a stick were extremely important notational devices. Our inborn number sense is limited to $7\pm 2$. Beyond nine, it's just "many". Those marks on a stick were important in letting our human ancestors herd sheep and cattle, knowing when to plant and when to harvest. Notational devices oftentimes are the key that lets mathematics and humanity advance to the next plateau. Zero? That's just a notational device, literally so in the case of the zero in 2014. Abstract symbols like "1", "4", and "8"? Those are nothing more than notational devices for |, ||||, and |||||||||. The symbols "+", "-", etc.: Those are just notational devices, too. We could be verbose and circumlocutious and use words and Roman numerals instead. Writing 2014+342 is just a bunch of notational devices.
Conceptually, we could do even complex mathematics using nothing but marks on a stick and the right words. Except we couldn't. Notational devices are important. They let us see the forest for the trees. In the case of matrices, I don't see how physicists could have developed quantum mechanics or how NASA could have gotten to the Moon without them. Quantum mechanics uses infinite dimension matrices. Writing that down using marks on a stick would take a long time. The Kalman filter, while not infinite dimension, is hairy in its own right. The matrix notation is what led to the development of to the signal processing concepts that in turn led to the Kalman filter.