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.
