Exercise · ZW-CENTER-OF-A-GROUP-004 · ★☆☆ Basic
What is the center of the group \(GL_2(\mathbb{R})\) of invertible \(2\times2\) matrices?
Choose one option.
Show answer
- A. The scalar matrices \(\lambda I\) with \(\lambda\neq 0\)
Explanation
A matrix that commutes with all invertible ones must be a scalar multiple of \(I\). A general diagonal matrix like \(\operatorname{diag}(1,2)\) does not commute with the matrix that swaps the rows.
Review the concept: Center of a group
The center of a group \(G\) is the set \(Z(G)\) of elements of \(G\) that commute with every element of \(G\).
\[Z(G)=\{\,z\in G : zg=gz\ \text{for all } g\in G\,\}\]
More practice problems: center of a group
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