MATH136 Lecture Notes - Invertible Matrix
29 Apr 2014
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MATH136 Full Course Notes
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Wednesday, march 12 lecture 26 : 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 k as a k = (a 1)k. it can be shown that, with this definition, the rules of exponents hold as usual for matrices. 26. 2 proposition suppose a1, a2, , ak are invertible matrices. Then: their product a1a2 ak is invertible, the inverse of the product of invertible matrices is the product oft he. Hence the product of invertible matrices is always invertible. (a1a2ak) 1 = ak. Proof : we outline the proof for the case where k = 3. And so b is the inverse of a1a2a3. 26. 3 proposition if a is an invertible matrix then [at ] 1 = [a 1]t. Then bat = [a 1]t at = [a a 1]t = i t = i (by a property of matrix algebra)
