Matrices and Determinants

If
$$\begin{vmatrix}x^2+x&x+1&x-2\\2x^2+3x-1&3x&3x-3\\x^2+2x+3&2x-1&2x-1\end{vmatrix}=ax-12,$$
then 'a' is equal to :

Let $M$ and $N$ be two $3\times3$ matrices such that $MN=NM$. Further, if $M\ne N^2$ and $M^2=N^4$, then

Let $M$ be a $2\times2$ symmetric matrix with integer entries. Then $M$ is invertible if

If $\alpha,\beta\ne0$, and $f(n)=\alpha^n+\beta^n$ and
$$\begin{vmatrix}3&1+f(1)&1+f(2)\\1+f(1)&1+f(2)&1+f(3)\\1+f(2)&1+f(3)&1+f(4)\end{vmatrix}=K(1-\alpha)^2(1-\beta)^2(\alpha-\beta)^2,$$
then K is equal to :

If A is an $3\times3$ non - singular matrix such that $AA'=A'A$ and $B=A^{-1}A'$, then $BB'$ equals :

If $a$, $b$, $c$ are non - zero real numbers and if the system of equations
$$(a-1)x=y+z,\qquad(b-1)y=z+x,\qquad(c-1)z=x+y,$$
has a non-trivial solution, then $ab+bc+ca$ equals :

If B is a $3\times3$ matrix such that $B^2=0$, then $\det\left[(I+B)^{50}-50B\right]$ is equal to :

Let A be a $3\times3$ matrix such that
$$A\begin{bmatrix}1&2&3\\0&2&3\\0&1&1\end{bmatrix}=\begin{bmatrix}0&0&1\\1&0&0\\0&1&0\end{bmatrix}.$$
Then $A^{-1}$ is :

Let for $i=1,2,3$, $p_i(x)$ be a polynomial of degree 2 in $x$, $p_i'(x)$ and $p_i''(x)$ be the first and second order derivatives of $p_i(x)$ respectively. Let,
$$A(x)=\begin{bmatrix}p_1(x)&p_1'(x)&p_1''(x)\\p_2(x)&p_2'(x)&p_2''(x)\\p_3(x)&p_3'(x)&p_3''(x)\end{bmatrix}$$
and $B(x)=[A(x)]^TA(x)$. Then determinant of $B(x)$ :

If
$$\begin{vmatrix}a^2&b^2&c^2\\(a+\lambda)^2&(b+\lambda)^2&(c+\lambda)^2\\(a-\lambda)^2&(b-\lambda)^2&(c-\lambda)^2\end{vmatrix}=k\lambda\begin{vmatrix}a^2&b^2&c^2\\a&b&c\\1&1&1\end{vmatrix},\ \lambda\ne0,$$
then k is equal to :

If $A=\begin{bmatrix}1&2&x\\3&-1&2\end{bmatrix}$ and $B=\begin{bmatrix}y\\x\\1\end{bmatrix}$ be such that $AB=\begin{bmatrix}6\\8\end{bmatrix}$, then :

If the three distinct lines $x+2ay+a=0$, $x+3by+b=0$ and $x+4ay+a=0$ are concurrent, then the point (a, b) lies on a :

If $f(\theta)=\begin{vmatrix}1&\cos\theta&1\\-\sin\theta&1&-\cos\theta\\-1&\sin\theta&1\end{vmatrix}$ and A and B are respectively the maximum and the minimum values of $f(\theta)$, then (A, B) is equal to :

Let A and B be any two 3×3 matrices. If A is symmetric and B is skewsymmetric, then the matrix AB − BA is :

If
$$\Delta_r=\begin{vmatrix}r&2r-1&3r-2\\\dfrac n2&n-1&a\\\dfrac12n(n-1)&(n-1)^2&\dfrac12(n-1)(3n+4)\end{vmatrix},$$
then the value of $\displaystyle\sum_{r=1}^{n-1}\Delta_r$ :

For $3\times3$ matrices $M$ and $N$, which of the following statement(s) is (are) NOT correct?

Let $\omega$ be a complex cube root of unity with $\omega\ne1$ and $P=[p_{ij}]$ be a $n\times n$ matrix with $p_{ij}=\omega^{i+j}$. Then $P^2\ne0$, when $n=$

Let $P=[a_{ij}]$ be a $3\times3$ matrix and let $Q=[b_{ij}]$, where $b_{ij}=2^{i+j}a_{ij}$ for $1\le i,j\le3$. If the determinant of $P$ is 2, then the determinant of the matrix $Q$ is

If $P$ is a $3\times3$ matrix such that $P^T=2P+I$, where $P^T$ is the transpose of $P$ and $I$ is the $3\times3$ identity matrix, then there exists a column matrix $X=\begin{bmatrix}x\\y\\z\end{bmatrix}\ne\begin{bmatrix}0\\0\\0\end{bmatrix}$ such that

If the adjoint of a $3\times3$ matrix $P$ is $\begin{bmatrix}1&4&4\\2&1&7\\1&1&3\end{bmatrix}$, then the possible value(s) of the determinant of $P$ is (are)