Here both A) and B) will be false..Let us see how ..For this we should know :
Determinant of LTM = Determinant of UTM = product of diagonal values which are also eigen values.
So considering this property , it may be possible that some diagonal element will be 0 and hence the determinant will be 0 and hence the matrix is singular..
Hence the matrix will be non invertible under multiplication operation..But we know ,
In group , inverse property must hold under the concerned operation..So both lower and diagonal matrices are not groups..To be precise they are monoids as they follow closure , associative and identity properties with identity matrix being the identity element if we consider a matrix as an element..
Option c) is correct as addition of matrix is done for corresponding elements , so every property of Abelian group is satisfied..Identity matrix will be NULL matrix in this case..And inverse matrix will contain all corresponding additive inverses of the elements of original matrix which is in other words are negatives..And commutative property also holds for addition.
Hence option C) is correct..
