If matrix P = $\begin{bmatrix$}[r] \text{cosec}^2 θ & 0 \ \text{tan}^2 θ & \text{cos}^2 θ \end{bmatrix}\text{ and matrix Q} = \begin{bmatrix}[r] -\text{cot}^2 θ & -1 \ -\text{sec}^2 θ & \text{sin}^2 θ \end{bmatrix}, find the matrix P + Q.

By formula,

sin² θ + cos² θ = 1,

cosec² θ - cot² θ = 1,

sec² θ - tan² θ = 1.

Substituting values in P + Q, we get :

$\Rightarrow \begin{bmatrix$}[r] \text{cosec}^2 θ & 0 \ \text{tan}^2 θ & \text{cos}^2 θ \end{bmatrix} + \begin{bmatrix}[r] -\text{cot}^2 θ & -1 \ -\text{sec}^2 θ & \text{sin}^2 θ \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] \text{cosec}^2 θ - \text{cot}^2 θ & 0 + (-1) \ \text{tan}^2 θ - \text{sec}^2 θ & \text{cos}^2 θ +\text{ sin}^2 θ \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] 1 & -1 \ -(\text{sec}^2 θ - \text{tan}^2 θ) & 1 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] 1 & -1 \ -1 & 1 \end{bmatrix}

Hence, P + Q = $\begin{bmatrix$}[r] 1 & -1 \ -1 & 1 \end{bmatrix}.