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Question

# 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. Statement-l: If A≠I and A≠−I, then detA=−1.Statement-2: If A≠I and A≠−I, then t1{A)≠0 .

A
Statement-1 is true, Statement-2 is true; Statement -2 is not a correct explanation for Statement-1.
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B
Statement-1 is true, Statement-2 is false.
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C
Statement-1 is false, Statement-2 is true.
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D
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
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Solution

## The correct option is B Statement-1 is true, Statement-2 is false.Let A=[abcd] where a,b,c,d∈Rtr(A)=a+dNow, A2=[abcd][abcd]⇒A2=[a2+bcab+bdac+cdbc+d2]Given A2=I⇒[a2+bcab+bdac+cdbc+d2]=[1001]⇒a2+bc=1=bc+d2 and ab+bd=0=ac+cdSince, A≠I and A≠−I⇒a=−dSo,a=√1−bc and d=−√1−bcHence, A=[√1−bcbc−√1−bc]⇒|A|=∣∣∣√1−bcbc−√1−bc∣∣∣⇒|A|=−1Statement 1 is true.Here, tr(A)=√1−bc−√1−bc=0So, statement 2 is false.

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