Row equivalence matrix

Dependencies:

  1. Row equivalence of matrices
  2. Elementary row operation is matrix pre-multiplication
  3. Matrix multiplication is associative
  4. Inverse of product

Let $A$ be row-equivalent to $B$. Then there exists an invertible matrix $R$ such that $B = RA$.

Proof

There exists a sequence of elementary row operations which transforms $A$ to $B$. Let $R_1, R_2, \ldots, R_p$ be the matrices associated with those operations. Let $R = R_pR_{p-1}\ldots R_2R_1$. Since each $R_i$ is invertible, $R$ is also invertible and $R^{-1} = (R_pR_{p-1}\ldots R_2R_1)^{-1} = R_1^{-1}R_2^{-1}\ldots R_p^{-1}$.

Since $B$ is obtained by applying the operations $R_1, R_2, \ldots, R_p$, \[ B = R_p(\ldots(R_2(R_1(A)))\ldots) = RA \] by associativity of matrix multiplication

Dependency for:

  1. RREF([A|I]) = [I|inv(A)] iff A is invertible
  2. Full-rank square matrix is invertible
  3. Equations with row equivalent matrices have the same solution set

Info:

Transitive dependencies:

  1. /sets-and-relations/equivalence-relation
  2. Group
  3. Ring
  4. Polynomial
  5. Integral Domain
  6. Comparing coefficients of a polynomial with disjoint variables
  7. Field
  8. Semiring
  9. Matrix
  10. Matrix multiplication is associative
  11. Elementary row operation
  12. Every elementary row operation has a unique inverse
  13. Row equivalence of matrices
  14. Identity matrix
  15. Inverse of a matrix
  16. Inverse of product
  17. Elementary row operation is matrix pre-multiplication