Let $\theta=\frac{\pi}{5}$ and $\mathrm{A}=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right]$. If $\mathrm{B}=\mathrm{A}+\mathrm{A}^{4}$, then $\operatorname{det}(\mathrm{B})$ :

JEE Main · 2020 · 06 Sep 2020 · Shift 2 · Matrices

Let

θ = \frac{π}{5}

and

A = [cos θ - sin θ sin θ cos θ]

. If

B = A + A^{4}

, then

det (B)

Answer: (D)

The matrix

A

is a rotation through

θ

, so

A^{4}

is a rotation through

4 θ

and

B = A + A^{4}

has equal diagonal entries

cos θ + cos 4 θ

with off-diagonal entries

\pm (sin θ + sin 4 θ)

. Its determinant is

(cos θ + cos 4 θ)^{2} + (sin θ + sin 4 θ)^{2} = 2 + 2 cos 3 θ

. With

θ = \frac{π}{5}

det (B) = 2 + 2 cos \frac{3 π}{5} = \frac{5 - 5}{2} \approx 1.38

, which lies in

(1, 2)

JEE Main · 2020 · 06 Sep 2020 · Shift 2 · Single correct option