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Question

If A=300220453, then which of the following is (are) true?

A
A is invertible
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B
tr(adj(adj(A)))=144
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C
tr(adj(adj(A)))=108
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D
|adj(A)| is less than 400
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Solution

The correct option is D |adj(A)| is less than 400
A=300220453
|A|=180
A is invertible.


We know that A(adjA)=|A|In
Replace A by adjA, we have
(adjA)(adj(adjA))=|(adjA)|In
(adjA)(adj(adjA))=|A|n1In
(adj(adjA))=|A|n2A
Since n=3, (order of matrix)
(adj(adjA))=|A|A=18A

tr(adj(adjA))=18tr(A)=18×8
tr(adj(adjA))=144


|adjA|=|A|2=182=324<400

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