Question

Let \(A=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)\) be a \(2 \times 2\) real matrix with \(\operatorname{det} A=1\). If the equation \(\operatorname{det}\left(A-\lambda I_{2}\right)=0\) has imaginary roots \(\left(I_{2}\right.\) be the Identity matrix of order 2), then
(A) \((a+d)^{2}<4\)
(B) \((a+d)^{2}=4\)
(C) \((a+d)^{2}>4\)
(D) \((a+d)^{2}=16\)

Answer 1

Ans: (A)
Hint : \(|\mathrm{A}|=0 \quad \therefore \mathrm{ad}-\mathrm{bc}=1\)
\(\left|A-\lambda I_{2}\right|=0\)
\(\left|\begin{array}{cc}a-\lambda & b \\ c & d-\lambda\end{array}\right|=0\)
\(\therefore a d-(a+d) \lambda+\lambda^{2}-b c=0\)
\(\lambda^{2}-(a+d) \lambda+1=0\)
\(\therefore(a+d)^{2}<4\)

Answer 2

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Answer 3

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