Made Easy Test Series 2019: Set theory & Algebra - Groups

Question

set of all possible diagonal matrix of order n

ans given monoid

my doubt-why it cannot have inverse??

Comments

no

---

Inverse of a diagonal matrix is obtained by taking reciprocal of the diagonal elements .

i.e if a diagonal element is a1 , in inverse it will be 1/a1.

But what if a1=0?

A diagonal matrix cannot have all elements as 0 , but it can have 0 elements right?

In that case , 1/0 is undefined.

---

yes i was also thinking the same now...

may be u are correct ...

anyways thanks

Answer 1

Zero matrix is also diagonal matrix and we know we can’t have inverse for the matrix for which determinant == 0

so for Zero matrix we can’t have inverse => G is not a group

But other Property like Closure , Associativity (Matrix multiplication is Associative) and identity element ( identity matrix is identity element for Matrix Multiplication ) is there so it is monoid