Multiplicative inverse

Definition

The multiplicative inverse of a number is another number which, when multiplied with the original number, results in 1. It is also known as the reciprocal.

5 Must Know Facts For Your Next Test

1. The multiplicative inverse of a non-zero real number $a$ is $\frac{1}{a}$.
2. For any matrix $A$, its multiplicative inverse $A^{-1}$ satisfies $A \cdot A^{-1} = I$, where $I$ is the identity matrix.
3. A square matrix has a multiplicative inverse if and only if it is non-singular (i.e., its determinant is non-zero).
4. The multiplicative inverse of a fraction $\frac{a}{b}$ is $\frac{b}{a}$, provided neither $a$ nor $b$ are zero.
5. In modular arithmetic, the multiplicative inverse of an integer $a$ modulo $n$ exists if and only if $a$ and $n$ are coprime.

Review Questions

• What is the multiplicative inverse of 7?
• How do you find the multiplicative inverse of a matrix?
• Why does a matrix need to be non-singular to have an inverse?

Related terms

Identity Matrix: A square matrix in which all elements of the principal diagonal are ones and all other elements are zeros.

Determinant: A scalar value that can be computed from the elements of a square matrix and encodes certain properties of the matrix.

Reciprocal: Another term for the multiplicative inverse; for any non-zero number \(x\), its reciprocal is \(\frac{1}{x}\).