Definition:Invertible Matrix
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Definition
Let $\struct {R, +, \circ}$ be a ring with unity.
Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Let $\mathbf A$ be an element of the ring of square matrices $\struct {\map {\MM_R} n, +, \times}$.
Then $\mathbf A$ is invertible if and only if:
- $\exists \mathbf B \in \struct {\map {\MM_R} n, +, \times}: \mathbf A \mathbf B = \mathbf I_n = \mathbf B \mathbf A$
where $\mathbf I_n$ denotes the unit matrix of order $n$.
Such a $\mathbf B$ is the inverse of $\mathbf A$.
It is usually denoted $\mathbf A^{-1}$.
Non-Invertible Matrix
Let $\mathbf A$ have no inverse.
Then $\mathbf A$ is referred to as non-invertible.
Also known as
An invertible matrix is called by some authors a non-singular matrix or regular matrix.
Also see
- Inverse of Matrix Product: if both $\mathbf A$ and $\mathbf B$ are invertible matrices, then so is $\mathbf A \mathbf B$, and its inverse is $\mathbf B^{-1} \mathbf A^{-1}$.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 29$. Matrices
- 1998: Richard Kaye and Robert Wilson: Linear Algebra ... (previous) ... (next): Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.3$ The inverse of a matrix
- 2008: David Joyner: Adventures in Group Theory (2nd ed.) ... (previous) ... (next): Chapter $2$: 'And you do addition?': $\S 2.2$: Functions on vectors: $\S 2.2.5$: Determinants
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): invertible matrix
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): non-singular