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Question

# Assertion :Let A be a 2×2 matrix with real entries. Let I be the 2×2 identity matrix. Denote by tr(A), the sum of diagonal entries of A. Assume that A2=I.If A≠I and A≠−I, then det(A)=−1. Reason: If A≠I and A≠−I, then tr(A)≠0.

A
Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
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B
Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
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C
Assertion is correct but Reason is incorrect
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D
Assertion is incorrect but Reason is correct
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Solution

## The correct option is C Assertion is correct but Reason is incorrectLet A=[abcd].Now, A2=I⇒det(A2)=1⇒(detA)2=1⇒detA=±1.Also,A2=I⇒A=A−1⇒[abcd]=1detA[d−b−ca]If det A=1, thena=d,b=−b,c=−c⇒a=d,b=c=0.In this case A=[a00a]. |A|=1⇒a2=1⇒a=±1∴A=I or A=−I. A contradiction.Thus, det(A)=−1∴[abcd]=−[d−b−ca]=[−dbc−a]⇒a=−d⇒tr(A)=a+d=0.∴ Statement-1: is true and statement-2 is false.Hence, option C.

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