In the matrix $A = \begin{bmatrix} 2 & 5 & 19 & 0 \\ 1 & 2 & 0 & 1 \\ 2 & 7 & \sqrt{3} & \sqrt{5} \end{bmatrix}$(i) The order of the matrix,(ii) The number of elements,(iii) Write the elements $a_{13}, a_{21}, a_{33}, a_{24}, a_{23}$.
Let $a \in \mathbb{R}$ and $A$ be a matrix of order $3 \times 3$ such that $\det(A) = -4$ and $A + I = \begin{bmatrix} 1 & a & 1 \\ 2 & 1 & 0 \\ a & 1 & 2 \end{bmatrix}$, where $I$ is the identity matrix of order $3 \times 3$. If $\det((a+1)\operatorname{adj}((a-1)A)) = \frac{2^m}{3^n}$, $m, n \in \{0, 1, 2, \ldots, 20\}$, then $m + n$ is equal to:
Let $T=\{\alpha_1,\alpha_2,\beta\}$ be the 3 distinct roots of $x^2+x-1=0$ (note: a quadratic has at most 2 roots; here the set $T$ includes specific values). For a $3\times 3$ matrix $M=(a_{ij})$, let $R_i=a_{i1}+a_{i2}+a_{i3}$ and $C_j=a_{1j}+a_{2j}+a_{3j}$. Match entries in List-I with List-II:
P) Number of $M$ with all entries in $T$ such that $R_i=C_j=0$ for all $i,j$
Q) Number of symmetric $M$ with all entries in $T$ such that $C_j=0$ for all $j$
R) Skew-symmetric $M$ with $a_{ij}\in T$ for $i>j$ — number of solutions to $M(x,y,z)^T=(-a,a,0)^T$
S) $M$ with all entries in $T$, $R_i=0$ for all $i$ — absolute value of $\det(M)$
List-II: 1)1, 2)12, 3)∞, 4)6, 5)0
**Paragraph:** Let $\alpha,\beta,\gamma$ be real roots of $x^3+ax^2+bx+c=0$ ($a,b,c\in\mathbb{R}$, $a,b\neq 0$). If the system $\alpha u+\beta v+\gamma w=0$, $\beta u+\gamma v+\alpha w=0$, $\gamma u+\alpha v+\beta w=0$ has non-trivial solution, then the minimum value of $x(3x+2a)+b+1$ is:
A) $5$\quad B) $3$\quad C) $1$\quad D) $0$
Let $T=\{\alpha_1,\alpha_2,\beta\}$ be the 3 distinct roots of $x^2+x-1=0$ (note: a quadratic has at most 2 roots; here the set $T$ includes specific values). For a $3\times 3$ matrix $M=(a_{ij})$, let $R_i=a_{i1}+a_{i2}+a_{i3}$ and $C_j=a_{1j}+a_{2j}+a_{3j}$. Match entries in List-I with List-II:
P) Number of $M$ with all entries in $T$ such that $R_i=C_j=0$ for all $i,j$
Q) Number of symmetric $M$ with all entries in $T$ such that $C_j=0$ for all $j$
R) Skew-symmetric $M$ with $a_{ij}\in T$ for $i>j$ — number of solutions to $M(x,y,z)^T=(-a,a,0)^T$
S) $M$ with all entries in $T$, $R_i=0$ for all $i$ — absolute value of $\det(M)$
List-II: 1)1, 2)12, 3)∞, 4)6, 5)0
**Paragraph:** Let $\alpha,\beta,\gamma$ be real roots of $x^3+ax^2+bx+c=0$ ($a,b,c\in\mathbb{R}$, $a,b\neq 0$). If the system $\alpha u+\beta v+\gamma w=0$, $\beta u+\gamma v+\alpha w=0$, $\gamma u+\alpha v+\beta w=0$ has non-trivial solution, then the minimum value of $x(3x+2a)+b+1$ is:
A) $5$\quad B) $3$\quad C) $1$\quad D) $0$