$$\begin{aligned} &\text { Since, } \cos 2 \theta=0\\ &\begin{array}{lc} \Rightarrow & \cos 2 \theta=\cos \frac{\pi}{2} \Rightarrow 2 \theta=\frac{\pi}{2} \\ \Rightarrow & \theta=\frac{\pi}{4} \\ \therefore & \sin \frac{\pi}{4}=\frac{1}{\sqrt{2}} \text { and } \cos \frac{\pi}{4}=\frac{1}{\sqrt{2}} \\ \therefore & \left|\begin{array}{ccc} 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \end{array}\right|^2 \end{array} \end{aligned}$$

$$\begin{aligned} &\text { Expanding along } R_1 \text {, }\\ &=\left[-\frac{1}{\sqrt{2}}\left(\frac{1}{2}\right)+\frac{1}{\sqrt{2}}\left(-\frac{1}{2}\right)\right]^2=\left[\frac{-2}{2 \sqrt{2}}\right]^2=\left(\frac{-1}{\sqrt{2}}\right)^2=\frac{1}{2} \end{aligned}$$

If $A$ is a matrix of order $3 \times 3$, then $\left(A^2\right)^{-1}=\left(A^{-1}\right)^2$.

If $A$ is a matrix of order $3 \times 3$, then $\left(A^2\right)^{-1}=\left(A^{-1}\right)^2$.

If $A$ is a matrix of order $3 \times 3$, then the number of minors in determinant of $A$ are 9 . [since, in a $3 \times 3$ matrix, there are 9 elements]

The sum of products of elements of any row with the cofactors of corresponding elements is equal to value of the determinant.

$$\text { Let } \Delta=\left|\begin{array}{lll} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}\right|$$

Expanding along $R_1$,

$$\begin{aligned} & \Delta=a_{11} A_{11}+a_{12} A_{12}+a_{13} A_{13} \\ &=\text { Sum of products of elements of } R_1 \text { with their } \\ & \text { corresponding cofactors } \end{aligned}$$