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
27.2.1.1 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,
27.2.1.2 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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