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Lecture

# Lecture 26.pdf

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University of Waterloo

Mathematics

MATH 136

Robert Sproule

Winter

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Wednesday, March 12 − Lecture 26 : Properties of invertible matrices.
Concepts:
Properties of invertible matrices.
26.1 Definition − The negative power of a matrix. If k is a positive integer, and A has an
inverse A −1, we define A as A = (A ) . It can be shown that, with this definition, the
rules of exponents hold as usual for matrices.
26.2 Proposition − Suppose A , A 1 ..2, A ark invertible matrices. Then:
1) Their product A A 1 2 A is knvertible.
2) The inverse of the product of invertible matrices is the product oft he
−1 −1 −1 −1
(A1 2....Ak) = A A k k-1 ....A1 .
Hence the product of invertible matrices is always invertible.
Proof : We outline the proof for the case where k = 3.
−1 −1 −1
- Suppose A , 1 , 2 ,a3e invertible. Let B= A 3 A2 A 1 .
- Then (A A1 2 3 = (A A A1 2 3 3−1A 2−1A1−1) = I
- Similarly B(A A1 2 3= (A 3−1A 2−1A 11)(A 1 2 3 = I
- And so B is the inverse of A A A . That is (A A A ) = A −1 −1A −1A −1.
1 2 3 1 2 3 3 2 1
T −1 −1 T
26.3 Proposition − If A is an invertible matrix then [A ] = [A ] .
Proof:
- Suppose A is invertible. Let B = [A −1] .
- Then BA = [A ] A = [A A ] = I = I −1 T T (By a property of matrix algebra)
- Then A B = A [A ] = [A A ] = I = I T T (By a property of matrix algebra)
T T −1 −1 T
- And so B is the inverse of A . That is [A ]

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