Math 2.0

Linear and multilinear algebra; matrix theory Proposition

Invertibility and the reduced row echelon form

Builds on Invertible matrixReduced row echelon formAugmented matrixGauss–Jordan elimination

Statement

Let AA be a square matrix. Then AA is invertible if and only if its reduced row echelon form is the identity matrix II, that is, A∼IA \sim I.

Moreover, when AA is invertible, row reduction finds the inverse. Form the augmented matrix [A∣I][A \mid I] and apply Gauss–Jordan elimination. The result is [A∣I]∼[I∣A−1].[A \mid I] \sim [I \mid A^{-1}].

Remarks

Each row operation is left multiplication by an invertible matrix. So reducing AA to II gives a product EE of such matrices with EA=IEA = I. Applying the same operations to II gives EI=E=A−1EI = E = A^{-1}.

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Augmented matrix, concept, no evidence levelAugmented matrixconceptGauss–Jordan elimination, concept, no evidence levelGauss–Jordan eliminat…conceptInvertible matrix, concept, no evidence levelInvertible matrixconceptReduced row echelon form, concept, no evidence levelReduced row echelon f…conceptInvertibility and the reduced row echelon form, statement, no evidence levelInvertibility and the…statement
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  1. Augmented matrix concept
  2. Gauss–Jordan elimination concept
  3. Invertible matrix concept
  4. Reduced row echelon form concept
  5. Invertibility and the reduced row echelon form statement

Further reading

  1. Web https://github.com/davidaustinm/ula/blob/b6ea263ef23703072b7007485bb7a2d89c5f399b/src/section3-1.xml#L551, src/section3-1.xml:551
About this page
Tag1ZK55FS2BF
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AboutInvertible matrix, Reduced row echelon form
UsesInvertible matrix, Reduced row echelon form, Augmented matrix, Gauss–Jordan elimination
Changed4 November 2026

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