Let , , and be square matrices with small, and define
(1)
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where is the identity matrix. Then the inverse of is approximately
(2)
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This can be seen by multiplying
(3)
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(4)
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(5)
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(6)
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Note that if we instead let , and look for an inverse of the form , we obtain
(7)
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(8)
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(9)
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(10)
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In order to eliminate the term, we require . However, then , so so there can be no inverse of this form.
The exact inverse of can be found as follows.
(11)
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so
(12)
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Using a general matrix inverse identity then gives
(13)
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