Example 10-4. For Aan n×nsquare matrix, show that (A−^1 )−^1 =A. That is,
show the inverse of an inverse matrix is again the original matrix A.
Solution Let B=A−^1 so that B−^1 = (A−^1 )−^1 , then by definition of an inverse matrix
one can write
AB =AA −^1 =I.
Right-multiply this equation on both sides by B−^1 to obtain
ABB −^1 =IB −^1 =B−^1.
Using the result that BB −^1 =Iand that AI =A, this last equation simplifies to
AI =A=B−^1 = (A−^1 )−^1
which establishes the result.
Example 10-5. Show that (AB )−^1 =B−^1 A−^1. That is, show the inverse of a
product of two matrices is the product of the inverses in the reverse order.
Solution By definition
(AB )−^1 (AB ) = I.
so that if one postmultiples both sides of this equation by B−^1 and simplifies the
results, one finds
(AB )−^1 (AB )B−^1 =IB −^1
(AB )−^1 A(BB −^1 ) = B−^1 , associative law
(AB )−^1 AI =B−^1
(AB )−^1 A=B−^1
Now postmultiply both sides of this last equation by A−^1 to obtain
(AB )−^1 AA −^1 =B−^1 A−^1
(AB )−^1 I=B−^1 A−^1
(AB )−^1 =B−^1 A−^1
which establishes the result.
Methods for calculating the inverse of a square matrix, if the inverse exists,
are developed in a later section. In this section, the emphasis is on definitions,
terminology and certain operational properties associated with square matrices.
