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Elementary Row And Column Operation

  1. Interchange row \(i\) with row \(j\)
  2. Multiply row \(i\) with non-zero constant
  3. Add a multiple of a row \(i\) to row \(j\)

Theorem 3.6

Let \(A\in M_{m\times n}(F)\) of rank \(r\). Then, \(r\leq m\), \(r\leq n\), and by means of elementary row and column operations \(A\) can be transformed into a matrix with \(r\times r\) diagonal matrix.

Definition

An \(n\times n\) elementary matrix is obtained by performing an elementary operation of \(I_n\)

Corollary

Let \(A\in M_{m\times n}(F)\) of rank \(r\). Then exists invertible matrices \(B\in M_{m\times m}(F)\) and \(C\in M_{n\times n}(F)\) st \(D=BAC\)

Theorem 3.4

Let \(A\in M_{m\times n}(F)\)

If \(P\in M_{m\times m}(F)\) and \(Q\in M_{n\times n}(F)\) are invertible matrices, then:

  1. \(\text{Rank}(A)=\text{Rank}(AQ)\)
  2. \(\text{Rank}(A)=\text{Rank}(PA)\)
  3. \(\text{Rank}(A)=\text{Rank}(PAQ)\)

Proposition

Elementary operations on \(A\) does not change the rank of \(A\)

Corollary

  1. \(\text{rank}(A)=\text{rank}(A^{-1})\)
  2. \(\text{rank}(A)=\text{dim}(\text{span of row vectors of A})\)

\(BAC=D\implies C^TA^TB^T=D^T\)

Inverse of a Matrix

\(\text{rank}(A)=n\iff A\) is invertible, \(A\in M_{n\times n}(F)\)

Corollary

Inverse of \(A\) is a product of elementary matrices

Augmented Matrix

\((A|B)\)

\(C(A|B)=(CA|CB)\)

If \(A\) is invertible, \((A|I)\rightarrow A^{-1}(A|I)\rightarrow (I|A^{-1})\)

Theorem 3.9

Let \(K\) be set of solutions of \(Ax=b\)

Let \(K_H\) be set of solutions of \(Ax=0\)

Let \(s\in K\)

\(K=s+K_H=\{s+k|k\in K_H\}\)