In linear algebra and geometry, the sophisticated prefer to speak of the advantages of coordinate free representations, but, when the rubber meets the road, they often "shut the doors and compute with matrices," as one wag put it.*
I was reminded of that by my current interest in tensor networks, where the action is precisely in matrices (and their higher rank analogs.
* Actual quote, from Irving Kaplansky, speaking of himself and Paul Halmos:
We share a philosophy about linear algebra: we think basis-free, we write basis-free, but when the chips are down we close the office door and compute with matrices like fury.
And a different opinion from Dieudonne:
There is hardly any theory which is more elementary [than linear algebra], in spite of the fact that generations of professors and textbook writers have obscured its simplicity by preposterous calculations with matrices.
Both via Mathoverflow