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Question

If A is a square matrix, then (adj A)1=adj(A1)=

A
A1
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B
A|A|2
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C
A|A|
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D
A|A|
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Solution

The correct option is C A|A|
(I) A1=adj A|A|
Replace A with adj(A), we get
(adj A)1=adj(adj A)|adj A| =|A|n2A|A|n1 =A|A|

(II) A1=adj A|A|
Replace A with A1, we get
(A1)1=adj A1|A1|A=|A|adj(A1)adj(A1)=A|A|

Hence, (adj A)1=adj(A1)=A|A|

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