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Home » Let be the set of four 4 th roots of unity. Prepare the composition table for multiplication on and show that (v) every element in A has its multiplicative inverse.
Let A=(1,-1, i,-i) be the set of four 4 th roots of unity. Prepare the composition table for multiplication on \mathrm{A} and show that (v) every element in A has its multiplicative inverse.
Binary Operations | CBSE | Class 12 | Exercise 3B | Maths | RS Aggarwal
Let A=(1,-1, i,-i) be the set of four 4 th roots of unity. Prepare the composition table for multiplication on \mathrm{A} and show that (v) every element in A has its multiplicative inverse.

(v)

For multiplicative inverse of every element of A, a^{*} b= e where a, b \in A.

\begin{array}{l} 1 \times b_{1}=1 \\ b_{1}=1 \\ -1 \times b_{2}=1 \end{array}

\begin{array}{l} \mathrm{b}_{2}=-1 \\ \mathrm{i} \times \mathrm{b}_{3}=1 \\ \mathrm{~b}_{3}=\frac{1}{\mathrm{i}} \Rightarrow \frac{1}{\mathrm{i}} \times \frac{\mathrm{i}}{\mathrm{i}}=\frac{\mathrm{i}}{\mathrm{i}^{2}} \Rightarrow \frac{\mathrm{i}}{-1}=-\mathrm{i} \\ -i \times \mathrm{b}_{4}=1 \\ \mathrm{~b}_{4}=\frac{1}{-\mathrm{i}} \Rightarrow \frac{1}{-\mathrm{i}} \times \frac{\mathrm{i}}{\mathrm{i}}=\frac{\mathrm{i}}{-\mathrm{i}^{2}} \Rightarrow \frac{\mathrm{i}}{-(-1)}=\mathrm{i} \end{array}

So, multiplicative inverse of \mathrm{A}=\{1,-1,-\mathrm{i}, \mathrm{i}\}

i

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