Answer to Question #116065 in Abstract Algebra for Joshua

Question

Let A=left−1,1,−i,iright,i=sqrt−1. Let the binary operation be multiplication, what is the inverse element of i?
a.\\(-i\\)
b.\\(i\\)
c.1
d.\\(-1\\)

Answer 1

Given $A= \{1,-1,i,-i\}$ and binary operation ia multiplication.

Identity of given set is 1 under binary operation multiplication because $a \times 1 = 1\times a = a \ \forall \ a\in A$.

Inverse of $a\in A$ is $b$ if $b \in A$ and $a\times b = b \times a = 1$.

So, Inverse of $i$ is $-i$ because $i \times (-i) = (-i) \times i = -i^2 = 1$ and also $-i \in A$.

So, Answer is option a) $-i$.