what does this mean?

if that means : A[ 0 1 ] = [ 1 1 ]

than you can see this:

matrix are just a way to write how linear operator affect on some vector...

so in this case [ 0 1 ] is vector ( 0, 1 )

A[ 0 1 ] = A( 0,1 ) = ( 1, 1 )

so... A[ 1 -1 ] = A( 1, -1 ) = { A is linear } = A( 1, 0 ) - A( 0, 1 ) = A( 1, 0 ) - ( 1, 1 )

and A( 1, -1 ) = ( 2, 3 ) ----> ( 2, 3 ) = A( 1, 0 ) - ( 1, 1 )

so A( 1, 0 ) = ( 3, 4 )

and you have:

A( 0, 1 ) = ( 1, 1 )

so A ( in canonical(?) basis ) =

3 1

4 -1