Page 56 - 35Linear Algebra

56 Systems of Linear Equations

How to find M −1

(M|I) ∼ (I|M −1 )

Much use is made of the fact that invertible matrices can be undone with
EROs. To begin with, since each elementary row operation has an inverse,

M = E −1 E 2 −1 · · · ,
1

while the inverse of M is

M −1 = · · · E 2 E 1 .

This is symbolically verified by

M −1 M = · · · E 2 E 1 E −1 E −1 · · · = · · · E 2 E −1 · · · = · · · = I .
1 2 2

Thus, if M is invertible, then M can be expressed as the product of EROs.
(The same is true for its inverse.) This has the feel of the fundamental
theorem of arithmetic (integers can be expressed as the product of primes)
or the fundamental theorem of algebra (polynomials can be expressed as the
product of [complex] first order polynomials); EROs are building blocks of
invertible matrices.

2.3.3 The Three Elementary Matrices

We now work toward concrete examples and applications. It is surprisingly
easy to translate between EROs and matrices that perform EROs. The
matrices corresponding to these kinds are close in form to the identity matrix:

• Row Swap: Identity matrix with two rows swapped.

• Scalar Multiplication: Identity matrix with one diagonal entry not 1.

• Row Sum: The identity matrix with one off-diagonal entry not 0.

Example 27 (Correspondences between EROs and their matrices)

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