Question

8. Let G be the group of all 2x2 non-singular matrices
over the reals. Then center of G, Z(G) is
(A)
CO)
(20
(B)
:a ER
(C)
:aeR
(D) None of these​

Answer 1

Answer: $\left[\begin{array}{cc}a&0\\0&a\end{array}\right]$ where a≠0, is the center of G.

Similar Question:  Let G be the group of all 2x2 non-singular matrices

over the reals. Then center of G, Z(G) is?

Given: We are given that G is a group of 2x2 non-singular matrices over the reals.

To Find: We need to find the center of G i.e., we need to find Z(G).

Step-by-Step Explanation: Firstly, we need to see the definition of general linear group and center of a group.

Definition of General Linear group: A set of nxn matrices whose determinant is non-zero and whose entries are from real numbers forms a group called General linear group of nxn matrices. It is represented as GL(n,R).

Definition of center of a group: The subset of all elements of a group which commute with every element of a group is called the center of a group. It is represented by Z(G).

Step 1: We know that non-singular matrices have a non-zero determinant. Therefore, the above given group is the group GL(2,R).

Step2: We know that only diagonal matrices commute with every other matrix. Therefore, center of GL(2,R) is the set of all diagonal matrices in GL(2,R).

The set of all diagonal matrices in GL(2,R) is $\left[\begin{array}{cc}a&0\\0&a\end{array}\right]$ where a≠0. (Because the element $\left[\begin{array}{cc}0&0\\0&0\end{array}\right]$ does not belong to GL(2,R))

To know more about Group Theory, refer to the links below:

https://brainly.in/question/7785558

https://brainly.in/question/45817

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