This preview shows page 1. Sign up to view the full content.
Unformatted text preview: times row 1 to row 3.
Interchange rows 2 and 3. Multiply row 2 by . You end up
with I3 . Outline Elementary Matrix Inverse of an Elementary Matrix The Inverse of a Matrix Properties of an Invertible Matrix
If A is an n × n matrix, then the following are equivalent.
(a) A is invertible.
(b) Ax = 0 has only the trivial solution.
(c) rref (A) = In .
(d) A = E1 E2 . . . Er for some elementary matrices E1 , E2 , . . . , Er .
(a) ⇒ (b):
(b) ⇒ (c):
(c) ⇒ (d):
(d) ⇒ (a): Outline Elementary Matrix Inverse of an Elementary Matrix The Inverse of a Matrix Row Equivalent Matrices If a matrix B can be obtained from a matrix A by a ﬁnite
sequence of elementary row operations, then A can be
obtained from B with the reverse sequence of the inverse
elementary row operations.
We say A and B are row equivale...
View Full Document