Let A, other than I or \(-I\), be a \(2 \times 2\) real matrix such that \(A^2=I\), I being the unit matrix. Let \(\mathrm{Tr}(A)\) be the sum of diagonal elements of A.
Statement-1: \(\mathrm{Tr}(A)=0\)
Statement-2: \(\det(A)=-1\)
Statement-1: \(\mathrm{Tr}(A)=0\)
Statement-2: \(\det(A)=-1\)
Show Hint
- Statement-1 is true; Statement-2 is false.
- Statement-1 is true; Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
- Statement-1 is true; Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
- Statement-1 is false; Statement-2 is true.
The Correct Option is B
Solution and Explanation
Using \(A^2=I\): \((1+d)I=tA\).
If \(t\neq 0\) then \(A=\dfrac{1+d}{t}I\) would be a scalar matrix, and \(A^2=I\) would force \(A=\pm I\), which is excluded. Hence \(t=0\), and then \(1+d=0\), i.e. \(d=-1\).
So both statements are true. They are two separate consequences of the same relation (the eigenvalues are +1 and -1); the determinant value is not the reason for the trace being zero.
Hence Statement-2 is true but is not the correct explanation of Statement-1.
Answer: (2) Statement-1 is true; Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
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