# MATA23H3 Lecture Notes - Lecture 9: Elementary Matrix, European Route E6

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3 Feb 2016

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Course

Professor

MATA23 - Lecture 9 - Invertible Matrices

Inverses of Elementary Matrices Continued

• Every elementary matrix is invertible, and its inverse is also an elementary matrix.

–Row operation on an identitiy I: interchange the ith row and the jth row

–I3=

100

010

001

–R2⇐⇒ R3:

100

001

010

=E3

–Row operation on an identitiy I: interchange the jth row and the ith row

–I3=

100

010

001

–R3⇐⇒ R2:

100

001

010

=E4

–E3E4=

100

010

001

=I3

–Row operation on an identitiy I:c×ith row added the jth row, then −c×ith row added

the jth row, c6= 0

–R1→R1+ 2R2:

120

010

001

=E5

–R1→R1+ (−2)R2:

1−2 0

010

001

=E6

–E5E6=

100

010

001

=I3

• Properties of invertible matrices:

1. Let Abe an invertible matrix.

(a) A−1is unique

(b) A−1is invertible, i.e. A(A−1)=(A−1)A=I

1