Suppose A, B are two 3×3 matrices such that A–1 exists. Then (A−B)A−1(A+B) is equal to

A
$(A + B) A^{- 1} (A - B)$
B
$A^{- 1} B + B^{2}$
C
$(I - B A B^{- 1}) (A - B)$
D
$(I + B A B^{- 1}) (A + B)$

Solution:

$\begin{array}{l} (A - B) A^{- 1} (A + B) \\ = (I - B A^{- 1}) (A + B) \\ = A + B - B A^{- 1} A - B A^{- 1} B \\ = A - B A^{- 1} B \\ = A + B - B A^{- 1} A - B A^{- 1} B \\ = (A + B) - B A^{- 1} (A + B) \\ = (A + B) (I - B A^{- 1}) \\ = (A + B) A^{- 1} (A - B) \end{array}$

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