A is an mxn matrix with reduced row echelon form R.Prove that if rank A
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The proof comes from the definition of reduced-row-echelon form. In order to get from the original form (A) to the reduced form (R), we multiply the matrix A by "elementary matrices," each of which is invertible. So, we have the following equality:
R = E_k * E_(k-1) * ... * E_3 * E_2 * E_1 * A
Where each E_i is an elementary matrix. We can rewrite this as
R = P * A
where P is the product of these Elementary matrices. If rank(A)
F*R = P*A
since F is invertible, we can write
R = F^-1 * P * A
Thus, (F^-1 * P) and P are two different invertible matrices that work.
QED
R = E_k * E_(k-1) * ... * E_3 * E_2 * E_1 * A
Where each E_i is an elementary matrix. We can rewrite this as
R = P * A
where P is the product of these Elementary matrices. If rank(A)
F*R = P*A
since F is invertible, we can write
R = F^-1 * P * A
Thus, (F^-1 * P) and P are two different invertible matrices that work.
QED
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i actually dont know
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keywords: Linear,Algebra,question,Linear Algebra question