# Nonsingular matrices and equivalences

Put content here**Nonsingular matrices** (also called **invertible** or **nondegenerate** matrices) are square matrices that have a multiplicative inverse. They are central to linear algebra because they represent bijective (one-to-one and onto) linear transformations.
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An $n \times n$ matrix $A$ is **nonsingular** if there exists $A^{-1}$ such that $AA^{-1} = A^{-1}A = I$.
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The **Invertible Matrix Theorem** (or Nonsingular Matrix Theorem) provides dozens of equivalent conditions for nonsingularity, connecting concepts from:
- Linear systems ($Ax = b$ has unique solutions)
- Vector spaces (columns/rows form a basis)
- Linear transformations (injectivity, surjectivity, isomorphism)
- Matrix algebra (existence of inverse, nonzero determinant)
- Rank and nullity (full rank, trivial null space)
- Eigenvalues (0 is not an eigenvalue)
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The child nodes of this node explore these equivalences in detail.

# Parents

* Matrices