I have a huge led matrix which will draw 3.1Amps of current. However I only have a 15v 1A psu> Would it be possible to use a transistor to amplify the current after I've lowered the voltage? Could you recommend such a transistor.
A transistor does not create current (and power) out of thin air. Transistors can amplify voltage or current but only if you provide them with an adequate supply.
A transistor is not magic at all. It works like a valve: it can pass more or less current (water) but if there is none available in the hose (supply) it will not be able to "create" it.
The amplification part comes from the ability to take small current or voltage and use it to control the transistor (valve) which can pass or block much higher currents and voltages.
If you only have 15W available, then the best switched mode power supply will get you between 1.1 and 1.25 amps at 12V, because switched mode power supplies can be 90% or more efficient. But not over 100% efficient, so for 12V at 3.5A you would need an initial supply of 15V at 2.8A at least. Alternatively, your 15W supply could be converted to a 3.5A at just less than 4V supply.
A transistor on its own will not perform this conversion but will be an important part of a switched mode power supply.