MATA23H3 Lecture Notes - Lecture 9: Elementary Matrix, European Route E6
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3 Feb 2016
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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
–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