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THEOREM 1.4.7
Laws of Exponents
If A is a square matrix and r and s are integers, then
The next theorem provides some useful properties of negative exponents.
THEOREM 1.4.8
Laws of Exponents .
If A is an invertible matrix, then:
(a) is invertible and
(b) is invertible and for .
(c) For any nonzero scalar k, the matrix is invertible and .
Proof , the matrix is invertible and .
(a) Since
(b) This part is left as an exercise.
(c) If k is any nonzero scalar, results (l) and (m) of Theorem 1.4.1 enable us to write
Similarly, so that is invertible and .
EXAMPLE 8 Powers of a Matrix
