Multiplicative inverse of a matrix

Definition

The multiplicative inverse of a matrix $A$ is another matrix $A^{-1}$ such that when $A$ is multiplied by $A^{-1}$, the result is the identity matrix. Not all matrices have a multiplicative inverse; only square matrices with non-zero determinants do.

5 Must Know Facts For Your Next Test

1. A matrix must be square (same number of rows and columns) to have a multiplicative inverse.
2. The determinant of the matrix must be non-zero for its inverse to exist.
3. $AA^{-1} = I$ and $A^{-1}A = I$, where $I$ is the identity matrix.
4. The formula to find the inverse of a 2x2 matrix $\begin{pmatrix}a & b\\ c & d\end{pmatrix}$ is $\frac{1}{ad - bc}\begin{pmatrix}d & -b\\ -c & a\end{pmatrix}$, provided that $ad - bc \neq 0$.
5. Inverting larger matrices often requires methods such as Gaussian elimination or using adjugates and cofactors.

Review Questions

• What conditions must be met for a matrix to have an inverse?
• Explain how you would find the inverse of a 3x3 matrix.
• What does it mean if multiplying two matrices results in the identity matrix?

Related terms

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

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

Gaussian Elimination: \textbackslash A method for solving linear systems of equations by transforming the system's augmented matrix into reduced row echelon form. It can also be used to find inverses of matrices.