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Unformatted text preview: eger and let A be an n × n matrix.
We use the following conventions:
A0 = In .
Ar = AA · · · A.
If A is invertible, then A−r = A−1 A−1 · · · A−1 . Outline Identity Matrices Inverse of a Matrix Properties of Matrix Powers
If A is an n × n matrix and r , s ∈ Z, then (assuming A is invertible
1 Ar As = Ar +s . 2 (Ar )s = Ars . 3 If A is invertible, then so is A−1 , and (A−1 )−1 = A. 4 If A is invertible, then so is Ar , and (Ar )−1 = A−r . 5 6 If A is invertible and k = 0, then aA is invertible, and
(kA)−1 = A−1 .
If A is invertible, then so is AT , and (AT )−1 = (A−1 )T . Proof.
Check. Outline Identity Matrices Inverse of a Matrix Examples Example
1 Find A−3 if
A= 2 11
12 If p (x ) = 2x 2 − x + 3 and
ﬁnd p (A). 11
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