Multiplicative inverse of a matrix

Definition

The multiplicative inverse of a matrix is another matrix that, when multiplied with the original matrix, yields the identity matrix. Only square matrices that are non-singular (having a non-zero determinant) have an inverse.

5 Must Know Facts For Your Next Test

1. A square matrix $A$ has an inverse if and only if its determinant $\text{det}(A) \neq 0$.
2. The product of a matrix $A$ and its inverse $A^{-1}$ is the identity matrix $I$, such that $AA^{-1} = I$.
3. Methods to find the inverse include using row reduction, the adjugate method, or applying elementary matrices.
4. For a 2x2 matrix $\begin{pmatrix}a & b\\c & d\end{pmatrix}$, its inverse is given by $\frac{1}{ad - bc}\begin{pmatrix}d & -b\\-c & a\end{pmatrix}$ provided $ad - bc \neq 0$.
5. Inverses are used in solving systems of linear equations by transforming them into simpler forms.

Review Questions

• What condition must be met for a square matrix to have an inverse?
• How do you verify that two matrices are multiplicative inverses of each other?
• What is one method to find the inverse of a given matrix?

Related terms

Determinant: A scalar value derived from a square matrix that determines whether the matrix has an inverse. It is zero for singular matrices.

Identity Matrix: A square diagonal matrix with ones on the main diagonal and zeros elsewhere. It acts as the multiplicative identity in matrix multiplication.

Elementary Matrix: Matrices obtained by performing a single elementary row operation on an identity matrix. Used in finding inverses and solving systems.