Multiplicative inverse

Definition

The multiplicative inverse of a number is another number that, when multiplied together, yield the product 1. For any nonzero number $a$, its multiplicative inverse is $\frac{1}{a}$.

5 Must Know Facts For Your Next Test

1. The multiplicative inverse of a matrix $A$ is denoted as $A^{-1}$, and it satisfies $A \cdot A^{-1} = I$, where $I$ is the identity matrix.
2. Only square matrices (matrices with the same number of rows and columns) can have a multiplicative inverse.
3. A matrix has an inverse if and only if its determinant is non-zero.
4. To find the inverse of a $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$, use the formula: $\frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$
5. The process for finding the inverse of larger matrices generally involves row reduction to achieve the identity matrix on one side.

Review Questions

• What condition must be satisfied for a square matrix to have an inverse?
• How would you find the multiplicative inverse of the number 7?
• What is the result when you multiply a matrix by its multiplicative inverse?

Related terms

Determinant: A scalar value that can be computed from the elements of a square matrix and determines whether a matrix has an inverse.

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

Row Reduction: :The process used to simplify matrices to row echelon form or reduced row echelon form, often used in finding inverses or solving systems of equations.