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

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set of all possible diagonal matrix of order n

ans given monoid

my doubt-why it cannot have inverse??

recategorized
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Can you guarantee that "Every Diagonal matrix" will be non-singular?
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a diagonal matrix is the one in which the prinicipal diagonal elements are non zero....so yes it is non singular
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Correct . Is there any other criteria given regarding matrix in the question?
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no
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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.
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yes i was also thinking the same now...

may be u are correct ...

anyways thanks

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

## Related questions

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Which of the following is true? Every lower triangular matrix is group under multiplication operation where all elements of diagonal are non zero numbers. Every diagonal matrix is group under multiplication operation, where all elements of diagonal are non zero numbers. Every matrix is Abelian under addition operation where all elements are real numbers. Both (a) and b) Why is C incorrect?