Exercise · ZW-INVERSE-ELEMENT-004 · ★★☆ Intermediate
Which identity is correct?
Choose one option.
Show answer
- A. (ab)⁻¹ = b⁻¹a⁻¹
Explanation
(ab)⁻¹ = b⁻¹a⁻¹: the order is reversed. Check: (ab)(b⁻¹a⁻¹) = a(bb⁻¹)a⁻¹ = aa⁻¹ = e. The version a⁻¹b⁻¹ only holds if the group is abelian.
Review the concept: Inverse element
Given \(a\in G\), an element \(b\in G\) is an inverse of \(a\) if operating in both orders gives the neutral element.
\[a\ast a^{-1}=a^{-1}\ast a=e\]
More practice problems: inverse element
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- ZW-INVERSE-ELEMENT-005 If a⁻¹ is the inverse of a, then...
- ZW-INVERSE-ELEMENT-006 The additive inverse of -5 is...
- ZW-INVERSE-ELEMENT-007 Which statement correctly describes the inverse of an element of a group?
- ZW-INVERSE-ELEMENT-008 Which condition guarantees that a square matrix has an inverse?
- ZW-INVERSE-ELEMENT-009 If b is the inverse of a, which equality always holds?