 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2013-resit/9/b/solution

The denominator for taking reciprocals a second time is

D^{'} = 1/ D

. The preceding cross-product result therefore gives

\frac{b ^{'} \times c ^{'}}{D ^{'}} = \frac{a / D}{1/ D} = a .

b

and

c

. Thus taking the reciprocal basis twice returns the original basis. The matrix statement is the same reciprocal basis involution:

(B^{- T})^{- T} = B

. Noncoplanarity is essential, since otherwise the defining denominator and the matrix inverse do not exist.

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