Question

Let $A$ be the set of all nonsingular matrices over real numbers and let $\ast$ be the matrix multiplication operator. Then

• A. $A$ is closed under $\ast$ but $<A,\ast >$ is not a semigroup
• B. $<A,\ast >$ is a semigroup but not a monoid
• C. $<A,\ast >$ is a monoid but not a group
• D. $<A,\ast >$ is a group but not an abelian group

Answer 1

The correct option is D $<A,\ast >$ is a group but not an abelian group

(i) Closure Property: Multiplication of two non-singular matrices is also non-singular matrix. Matrix multiplication over non-singular matrices follows closure properties.

(ii) Associative Property: Multiplication over any set of matrices is associative.
(AB)C = A(BC)
Where A, B and C are non-singular matrices

(iii) Identity Element: Identity matrix I is the identity element for matrix multiplication over matrices and 1 is non-singular

(iv) Inverse Element: For every non-singular matrix its inverse exists. So, for non-singular matrices inverse element exist.

(v) Commutative: Matrix multiplication is not commutative
$AB\ne BA$
Where $A,B$ are non-singular matrices.
Matrices multiplication is not commutative.
So $<A,\ast >$ is a group but not an abelian group.