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. A square matrix $A$ is invertible if there exists a matrix $B$ such that $AB = BA = I$, where $I$ is the identity matrix.
2. The inverse of a 2x2 matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is given by $\frac{1}{ad - bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$, provided that $ad - bc \neq 0$.
3. For an $n \times n$ matrix to be invertible, its determinant must be non-zero ($det(A) \neq 0$).
4. If a matrix can be row reduced to the identity matrix using elementary row operations, it is invertible.
5. The product of two invertible matrices is also an invertible matrix.

Review Questions

• What condition must the determinant of a square matrix meet for it to be invertible?
• How can you find the inverse of a 2x2 matrix?
• What operation confirms that a given square matrix is indeed invertible?

Related terms

Determinant: A scalar value derived from a square matrix that indicates whether the matrix is invertible and encodes certain properties of linear transformations represented by the matrix.

Identity Matrix: A square matrix in which all elements of the principal diagonal are ones and all other elements are zeros. It acts as the multiplicative identity in matrix multiplication.

Elementary Row Operations: Operations including row swapping, scaling rows, and adding/subtracting multiples of rows used in solving systems of linear equations and finding inverses.