Theorem: Let A be nxn matrix on real numbers, then Det(Aᵗ)=Det(A) Proof: Let e be any elementary row operation on nxn matrices. Then, define e’ to be analogous column operation on nxn matrices. What do we mean by the word “analogous” here? Well, let’s say e multiplies r’th row by c and adds it to …
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Every matrix is row equivalent to a single row-reduced echelon matrix
If AB=I, then BA=I
Proposition: Let A and B be nxn matrices. Also, let AB=I then BA=I Proof: Every matrix is row equivalent to a single row-reduced echelon matrix. That is, for every matrix, there is a finite sequence of elementary row operations such that if we apply this sequence of row operations on this matrix, we get a …
Two equivalent systems of linear equations’ augmented matrices are row equivalent
Proposition: Let A and B mxn matrices. Also, let AX=Y and BX=Y be equivalent systems of linear equations. Then, A’=B’ where