Invertible matrix

Definition

An invertible matrix is a square matrix that has an inverse, meaning there exists another matrix which when multiplied with the original yields the identity matrix. A matrix is invertible if and only if its determinant is non-zero.

5 Must Know Facts For Your Next Test

1. An $n \times n$ matrix $A$ is invertible if there exists an $n \times n$ matrix $B$ such that $AB = BA = I$, where $I$ is the identity matrix.
2. The determinant of an invertible matrix is non-zero: $\det(A) \neq 0$.
3. If a matrix $A$ is invertible, then its inverse is unique and denoted as $A^{-1}$.
4. The product of two invertible matrices is also invertible, and $(AB)^{-1} = B^{-1}A^{-1}$.
5. A system of linear equations represented by $AX = B$ has a unique solution if $A$ is invertible; the solution can be found using $X = A^{-1}B$.

Review Questions

• What condition must the determinant of a matrix satisfy for it to be considered invertible?
• If $A$ and $B$ are both invertible matrices, what can be said about the product $AB$? What is the relationship between their inverses?
• How can you use the inverse of a matrix to solve a system of linear equations?

Related terms

Determinant: A scalar value that can be computed from the elements of a square matrix. It provides important properties about the matrix, including whether it is invertible.

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

Cramer's Rule: A method used to solve systems of linear equations with as many equations as unknowns, employing determinants to find solutions.