linear algebra

Question

Consider the set H of all 3 × 3 matrices of the type:
a f e

0 b d

0 0 c
where a, b, c, d, e and f are real numbers and abc ≠ 0. Under the matrix multiplication operation,
the set H is:
(a) a group

(b) a monoid but not a group
(c) a semigroup but not a monoid

(d) neither a group nor a semigroup

Answer 1

Firstly check for Algebraic structure then Semi-group then Monoid and finally Group

Algebraic Structure:As it is Upper diagonal matrix so if we choose any two upper diagonal matrix and multiply them then we aggain get upper diagonal matrix so it closed under opeartion multiplication so it is Algebraic Structure

Semigroup:As we know that matrix multiplication is associative so it is Semi-group also

Monoid:for this there exist any identity element whose multiplication with any matrix from this set we again get same matrix

As identity matrix I_3x3 matrix exist so it is Monoid also

Group:for this  there exist inverse of every matrix

we know that condition for inverse is det(A) does not equal to zero

det(A) is abc (as in Upper diagonal matrix det is multiplication of diagonal elements) and it is given that abc does not equal to zero hence inverse for matrix of set H  exist Hence it is Group also

so answer is option A