MATH 232 2019 summer Lecture notes, Lecture notes of Mathematics

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Powers of a matrix If A is a square matrix, we define the non-negative integer powers of A to be et tad APS oA n factors and if A is invertible, then we define the negative powers of A to be Ane = (Att ae Acie Aes n factors Property If rand s are integers, then AvA? = ar and #¢a(AQ)y— Ae Theorem 3.2.9 If A is invertible and n is a non-negative integer, then 1. A~l is invertible and (A~!)-1=A 2. A” is invertible and (A")-! = A“ = (A-1)" 3. If k is a non-zero scalar, then (kA)~! = ja} Example. Expand (A + B)? (AtBY < (AtB)(AtB) = AA + AB+BATKE a ee Abt BA ee cunneh simplify AB+BR +o ZAR — becuse in general» ABEDA