MadeEasy Test Series: Set Theory & Algebra - Groups

Question

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?

Comments

yes all are correct

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given answer is correct .

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why is C wrong? @kunal chalotra

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@kunal. Could you share why its not Abelian? Even I am getting all options correct.

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closed+ associative + identity + inverse
i think inverse may not exit for matrix  because some matrix may have determinat 0. ??

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inverse of a matrix A with respect to addition operation is "-A", as identity matrix is null matrix.so inverse exists for every matrix when operation is addition. isnt it?

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@anusha. You are right. Find inverse with respect to addition operation.

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yes anusha .u r rt . i should think with respect to addition which intially i was not thinking . thanx .

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@kunal. You were taking multiplicative inverse

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@gabbar its addition not multiplication..here identity element is 0 and inverse is I(A)= -A

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@MADE EASY people are shit

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C is also correct. It it was asked for multiplication it would be false, as it can be singular.

Answer 1

A) Every lower triangular matrix is group under multiplication operation where all elements of diagonal are non zero numbers.

B)Every diagonal matrix is group under multiplication operation, where all elements of diagonal are non zero numbers.

Both are TRUE because Determinant of upper or lower triangular matrix and diagonal matrix is equal to the product of its diagonal elements. As it is given that diagonal elements are non zero it means Determinant of matrix is non zero which in turn means inverse of matrix exists.

As it is monoid and As Inverse of matrix also exists, Hence it is Group.

C) Every matrix is Abelian under addition operation where all elements are real numbers.

It is FALSE because it is not mentioned that Set of matrix is Addition compatible. it is not even closed.