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

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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
R2R3:
100
001
010
=E3
Row operation on an identitiy I: interchange the jth row and the ith row
I3=
100
010
001
R3R2:
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
R1R1+ 2R2:
120
010
001
=E5
R1R1+ (2)R2:
12 0
010
001
=E6
E5E6=
100
010
001
=I3
Properties of invertible matrices:
1. Let Abe an invertible matrix.
(a) A1is unique
(b) A1is invertible, i.e. A(A1)=(A1)A=I
1
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