Inverse matrix

Definition

An inverse matrix is a matrix that, when multiplied by the original matrix, results in the identity matrix. Only square matrices with a non-zero determinant have inverses.

5 Must Know Facts For Your Next Test

1. A matrix $A$ has an inverse if and only if it is square and its determinant is non-zero.
2. The inverse of a matrix $A$ is denoted as $A^{-1}$.
3. Multiplying a matrix by its inverse yields the identity matrix: $AA^{-1} = I$ and $A^{-1}A = I$.
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. To solve a system of linear equations using an inverse matrix, you can use the formula $X = A^{-1}B$, where $A$ is the coefficient matrix, $X$ is the variable matrix, and $B$ is the constant matrix.

Review Questions

• What conditions must be met for a square matrix to have an inverse?
• How do you denote the inverse of a given matrix $A$?
• What result do you get when you multiply a matrix by its inverse?

Related terms

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

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

$AX = B$: Represents a system of linear equations in which A is the coefficient matrix, X is the variable vector, and B is the constant vector.