If the matrix $A=\begin{bmatrix} 0 & 2 \\ K & -1 \end{bmatrix}$ satisfies $A(A^{3}+3I)=2I$,then the value of $K$ is:

Let $P = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$ be a matrix. Three elements of this matrix $P$ are selected at random. $A$ is the event of having the three elements whose sum is odd. $B$ is the event of selecting the three elements which are in a row or column. Then $P(A) + P(A|B) =$?

Let $A$ and $B$ be two invertible matrices of order $3 \times 3$. If $\det(ABA^T) = 8$ and $\det(AB^{-1}) = 8$,then $\det(BA^{-1}B^T)$ is equal to

$A$ square matrix $P$ satisfies $P^2 = I - P$. If $P^n = 5I - 8P$,then $n$ is:

$A = [a_{ij}]$ is a $3 \times 3$ matrix with positive integers as its elements. Elements of $A$ are such that the sum of all the elements of each row is equal to $6$ and $a_{22} = 2$. If $a_{ii} = \begin{cases} a_{ij} + a_{ji}, & j = i + 1 \text{ when } i < 3 \\ a_{ij} + a_{ji}, & j = 4 - i \text{ when } i = 3 \end{cases}$ for $i = 1, 2, 3$, then $|A| = $

Consider the following linear equations:
$ax+by+cz=0$,$bx+cy+az=0$,$cx+ay+bz=0$
Match the conditions/expressions in Column $I$ with statements in Column $II$:

Column $I$	Column $II$
$(A)$ $a+b+c \neq 0$ and $a^2+b^2+c^2=ab+bc+ca$	$(p)$ The equations represent planes meeting only at a single point.
$(B)$ $a+b+c=0$ and $a^2+b^2+c^2 \neq ab+bc+ca$	$(q)$ The equations represent the line $x=y=z$.
$(C)$ $a+b+c \neq 0$ and $a^2+b^2+c^2 \neq ab+bc+ca$	$(r)$ The equations represent identical planes.
$(D)$ $a+b+c=0$ and $a^2+b^2+c^2=ab+bc+ca$	$(s)$ The equations represent the whole of the three-dimensional space.