Question

Consider the set $H$ of all $3\times 3$ matrices of the type
$\left[\begin{array}{ccc}a& f& e\\ 0& b& d\\ 0& 0& c\end{array}\right]$
where $a,b,c,d,e$ and $f$ are real numbers and $abc\ne 0$. Under the matrix multiplication operation, the set $H$ is

• A. A group
• B. A monoid but not group
• C. A semigroup but not a monoid
• D. Neither a group nor a semigroup

Answer 1

The correct option is A A group

(i) The set $H$ is closed, since multiplication of upper triangular matrices will result only in upper traingular matrix.

(ii) Matrix multiplication is associative, i.e.,
$A\ast (B\ast C)=(A\ast B)\ast C$

(iii) Identify element is
$I=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$
and this belongs to $H$ as $I$ is an upper triangular as well as lower triangular matrix.

(iv) If $AϵH$, then $|A|=abc$. Since it is given that $abc\ne 0$, this means that $|A|\ne 0$ i.e. every matrix belonging to $H$
is non-singular and has a unique inverse.
$\therefore$ the set $H$ along with matrix multiplication is a group.