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# Equivalence theorem for nonsingular matrices: the matrix A row-reduces to the identity matrix.

**Theorem:** An  	imes $n \times n$ matrix $A$ is nonsingular if and only if The matrixiff $A$ row-reduces to the identity matrix $I_n$. This means $A$ is a product of elementary matrices (each corresponding to a row operation), and products of, hence invertible matrices are invertible.

# Parents

* Nonsingular matrices and equivalences