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Lecture 27.pdf

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MATH 136
Robert Sproule

Friday, March 14 − Lecture 27 : Elementary matrices and invertibility. Concepts: 1. Define row-equivalent matrices. 2. Recognize that a square matrixA is invertible iff A is row-equivalent to I n 3. Define elementary matrix. 4. Recognize that for any two row-equivalent matrices Aand B, there exist and invertible matrix P such that A = PB. 27.0 Recall − An ERO can always be reversed by an ERO. - The ERO P can be undone by applying P again. ij ij - The ERO cR caj be undone by applying (1/c)R . j - The ERO cR + i caj be undone by applying −cR + R i j 27.1 Definition − Row-equivalent matrices. Two matrices A and B are row-equivalent if there is a finite sequence of elementary row operations which, when applied successively, transform the matrix A into the matrix B. We will use the notation A ~ B to say “A is row-equivalent to B”. 27.1.1 Example − The following chain of ERO's applied to A shows that A is row- equivalent to B. 27.1.2 Observation − The matrix A is row-equivalent to B if and only if A RREF = BRREF. Proof : ( ⇒ ) If A ~ B, ERO’s ERO’s ERO’s “A → → B → → B RREF” ⇒ “A → → B RREF = A RREF” ( ⇐ ) If BRREF = ARREF, ERO’s ERO’s ERO’s “A → → A RREF = BRREF → → B” ⇒ “A → → B” and so A ~ B. For example, if then 27.2 Definition − An n × n elementary matrix is an n × n square matrix E n × nobtained by applying a single elementary row operation to I n × n 27.2.1 There are 3 types of elementary matrices: E , E , E P ij cRi cRi + Rj Each of these is obtained by applying one of the 3 elementary row operations of type I, II, or II respectively to I . n Verify this fact: Multiplying a matrix A m × non the left by an elementary matrix of type E produces the same matrix as the one obtained by applying an i elementary row operation of type i to A. That is, - if A → P → ijthen E A = B, Pij - if A → cR → Bithen E cR i = B, - if A → cR + Ri→ B tjen E cRi + Rj = B, Example − Compare the effect of the ERO, 4R + R , on the matr2x A t3 the result obtained by multiplying A on the left by E 4R2+ R3 27.3 Recall how an ERO can be“undone” by an ERO. (i.e. P undoes P , (1/c)R ij ij i undoes cR andi−cR + R undies cj + R. We veriiy thatjeach elementary matrix is invertible : −1 - E cRi = E (1/c)Ri −1 - E P 23 = E P 23
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