Multiplicative Inverse

In a monoid or multiplicative group where the operation is a product , the multiplicative inverse of any element  is the element  such that , with 1 the identity element.

The multiplicative inverse of a nonzero number  is its reciprocal  (zero is not invertible). For complex ,

The inverse of a nonzero real quaternion  (where  are real numbers, and not all of them are zero) is its reciprocal

where .

The multiplicative inverse of a nonsingular matrix is its matrix inverse.

To detect the multiplicative inverse of a given element in the multiplication table of finite multiplicative group, traverse the element's row until the identity element 1 is encountered, and then go up to the top row. In this way, it can be immediately determined that  is the multiplicative inverse of  in the multiplicative group  formed by all complex fourth roots of unity.