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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 1: If AI and AI, then detA=1
Statement 2: If AI and AI, then tr(A)0, then which of the following is correct

A
Statment 1 is false, statement 2 is true.
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B
Both statement 1 & 2 are true and statement 2 is a correct explanation for statement 1
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C
Both statement 1 & 2 are true but statement 2 is not a correct explanation for statement 1.
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D
Statement 1 is true, statement 2 is false
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Solution

The correct option is D Statement 1 is true, statement 2 is false
Let A=[abcd].
Then A2=[a2+bcab+bdac+dcbc+d2]=[1001]
Hence
det(A)=1bcbc1bc=1+bcbc=1
Here, we can see Tr(A)=0 so, 2nd statement is false only.

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