Multiplicative Inverse

The multiplicative inverse of a number is another number that, when multiplied with the original, gives the multiplicative identity, which is 1. In simpler terms, it is the number’s reciprocal.

Formal Definition

For any nonzero number \( a \), its multiplicative inverse is a number \( a^{-1} \) (or \(\frac{1}{a}\)) such that:

Examples

1. For a Real Number:

   • If \( a = 5 \), its multiplicative inverse is \( \frac{1}{5} \) because: \[ 5 \times \frac{1}{5} = 1. \]
   • If \( a = \frac{3}{4} \), then its multiplicative inverse is \( \frac{4}{3} \) because: \[ \frac{3}{4} \times \frac{4}{3} = 1. \]

2. For a Matrix:

   • If \( A \) is an invertible (non-singular) matrix, its multiplicative inverse \( A^{-1} \) is defined such that: \[ A \times A^{-1} = A^{-1} \times A = I, \] where \( I \) is the identity matrix.

3. In Modular Arithmetic:

   • Consider \( a \) modulo \( n \). The multiplicative inverse of \( a \) modulo \( n \) is a number \( b \) such that: \[ a \times b \equiv 1 \pmod{n}. \]
   • An inverse exists if and only if \( a \) and \( n \) are relatively prime (i.e., \(\gcd(a, n) = 1\)).

Notes

• Zero Exception: The number 0 does not have a multiplicative inverse because no number multiplied by 0 can produce 1.
• Usefulness: Multiplicative inverses are essential for solving equations, particularly in algebra and calculus, as well as for operations in more abstract algebraic structures like fields and rings.

In summary, the multiplicative inverse is a key concept that allows us to “undo” multiplication, much like subtraction undoes addition.

Latex

a \times a^{-1} = a^{-1} \times a = 1.
a \times \frac{1}{a} = \frac{1}{a} \times a = 1.