Let $A = \begin{bmatrix} m & n \\ p & q \end{bmatrix}$,$d = |A| \neq 0$ and $|A - d(\operatorname{Adj} A)| = 0$. Then:

Let $\omega$ be a complex cube root of unity with $\omega \neq 1$ and $P = [p_{ij}]$ be an $n \times n$ matrix with $p_{ij} = \omega^{i+j}$. Then $P^2 \neq 0$ when $n =$

Let $P=\left[\begin{array}{cc}\frac{\sqrt{3}}{2} & \frac{1}{2} \\ -\frac{1}{2} & \frac{\sqrt{3}}{2}\end{array}\right]$,$A=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]$ and $Q=PAP^{T}$. If $P^{T}Q^{2007}P=\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]$,then $2a+b-3c-4d$ is equal to:

Which of the following is incorrect?

Let $A$ and $B$ be two symmetric matrices of order $3$.
Statement $-1$: $A(BA)$ and $(AB)A$ are symmetric matrices.
Statement $-2$: $AB$ is a symmetric matrix if the matrix multiplication of $A$ with $B$ is commutative.

$\det \left[ \begin{array}{ccc} \frac{a^2+b^2}{c} & c & c \\ a & \frac{b^2+c^2}{a} & a \\ b & b & \frac{c^2+a^2}{b} \end{array} \right] = $