Consider the set H of all 3 x 3 matrices of the type ( a f e ) ( 0 b d ) ( 0 0 c ) where a, b, c, d, e, f are real numbers and a*b*c != 0. Under the matrix multiplication operation, the set H is: A. a group B. a monoid but not a group C. a semi group but not a monoid D. neither a group nor a semi group
GATE 2005 · Discrete Mathematics · Group Theory · medium
Answer: A. a group
- Check closure and associativity: Product of two upper-triangular matrices is upper-triangular. Diagonal entries of the product are products of diagonal entries, so nonzero (since a,b,c != 0 for both factors). Matrix multiplication is always associative. Closure and associativity hold.
- Check identity and inverses: Identity I_3 has diagonal (1,1,1), product 1 != 0, so I_3 is in H. For any A in H, det(A) = a*b*c != 0, so A is invertible. The inverse of an upper-triangular matrix with nonzero diagonal is upper-triangular with nonzero diagonal, so A^{-1} is in H.