Inverse Property

The inverse property involves using an opposite value or reciprocal to undo an operation. An inverse is heavily tied down to the concept of the identity element.

Definition

An element has an inverse with respect to an operation if combining the element and its inverse yields the identity element of that operation.

This is read as: For every element a in the set of real numbers (ℝ), there exists an element b in the same set such that a ∗ b = b ∗ a = e, where e is the identity element.

In simpler terms, this means that each element a has an inverse(denoted by b) that “undoes” the effect of the operation, bringing us back to the identity e.

In simpler and more specific terms, an inverse is also defined in terms of an operation and a set which can be seen as;

Examples

• $5 + (-5) = 0$ — Additive inverse
• $3 \times \frac{1}{3} = 1$ — Multiplicative inverse
• $A \times A^{-1} = I$ — Inverse of a square matrix

Special Cases

Not every element has an inverse. For example, 0 does not have a multiplicative inverse. Some matrices also lack an inverse (non-invertible or singular matrices). Other times, an inverse might exist but it might not be present in the set in question. For example, for Natural Numbers, while the multiplicative identity exists, their multiplicative inverse does not exist.

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