MATL, 17 bytes

4:"Gas2YLG@X!X=va

Try it online! Or verify all test cases.

Outputs 1 for Pascal matrices, 0 otherwise.

###Explanation

4: % Push [1 2 3 4]
" % For each
 G % Push input: ×ばつN
 a % ×ばつN vector containing 1 for matrix columns that have at least a nonzero
 % entry, and 0 otherwise. So it gives a vector containing 1 in all entries
 s % Sum. Gives N
 2YL % Pascal matrix of that size
 G % Push input
 @ % Push current iteration index
 X! % Rotate the matrix that many times in steps of 90 degress
 X= % Are they equal?
 v % Concatenate with previous accumulated result
 a % Gives 1 if at least one entry of the vector is nonzero
 % End (implicit). Display (implicit)

MATL, 17 bytes

4:"Gas2YLG@X!X=va

Try it online! Or verify all test cases.

Outputs 1 for Pascal matrices, 0 otherwise.

Explanation

4: % Push [1 2 3 4]
" % For each
 G % Push input: ×ばつN
 a % ×ばつN vector containing 1 for matrix columns that have at least a nonzero
 % entry, and 0 otherwise. So it gives a vector containing 1 in all entries
 s % Sum. Gives N
 2YL % Pascal matrix of that size
 G % Push input
 @ % Push current iteration index
 X! % Rotate the matrix that many times in steps of 90 degress
 X= % Are they equal?
 v % Concatenate with previous accumulated result
 a % Gives 1 if at least one entry of the vector is nonzero
 % End (implicit). Display (implicit)

MATL, 17 bytes

4:"Gas2YLG@X!X=va

Try it online! Or verify all test cases.

Outputs 1 for Pascal matrices, 0 otherwise.

MATL, 17 bytes

4:"Gas2YLG@X!X=va

Try it online! Or verify all test cases.

Outputs 1 for Pascal matrices, 0 otherwise.

###Explanation

4: % Push [1 2 3 4]
" % For each
 G % Push input: ×ばつN
 a % ×ばつN vector containing 1 for matrix columns that have at least a nonzero
 % entry, and 0 otherwise. So it gives a vector containing 1 in all entries
 s % Sum. Gives N
 2YL % Pascal matrix of that size
 G % Push input
 @ % Push current iteration index
 X! % Rotate the matrix that many times in steps of 90 degress
 X= % Are they equal?
 v % Concatenate with previous accumulated result
 a % Gives 1 if at least one entry of the vector is nonzero
 % End (implicit). Display (implicit)