Consider the following statements.Statement-1 : If $\begin{bmatrix} 3 & -2 \\ 3 & 0 \\ 2 & 4 \end{bmatrix} \begin{bmatrix} y & y \\ x & x \end{bmatrix} = \begin{bmatrix} 3 & 3 \\ 3y & 3y \\ 10 & 10 \end{bmatrix}$, then $2x + 3y = \lambda$.Statement-2 : Given that $\ell + 5 = p + 2m$, where $A$ is a square matrix of order $n$.$\ell$ = maximum number of distinct entries if $A$ is a triangular matrix.$m$ = maximum number of distinct entries if $A$ is a diagonal matrix.$p$ = minimum number of zeroes if $A$ is a triangular matrix.Statement-3 : Let $A$ be the set of all $3 \times 3$ skew symmetric matrices whose entries are either $-1, 0$ or $1$. If there are exactly three $0$'s, three $1$'s and three $(-1)$'s, then number of such matrices is equal to $\mu$.Then, which of the following options is/are correct?
If \(a_1, a_2, \ldots, a_n, \ldots\) form a G.P. and \(a_i > 0\), for all \(i \geq 1\), then \(\begin{vmatrix} \log a_n & \log a_{n+1} & \log a_{n+2} \\ \log a_{n+3} & \log a_{n+4} & \log a_{n+5} \\ \log a_{n+6} & \log a_{n+7} & \log a_{n+8} \end{vmatrix}\) is equal to
Which of the following options is/are correct ?(A) Let $A = \begin{bmatrix} 1 & 3 \\ -2 & 2 \end{bmatrix}$, $B = \begin{bmatrix} 4 & -3 \\ 2 & 2 \end{bmatrix}$ and $C_r = \begin{bmatrix} r.3^r & 2^r \\ 0 & (r-1)3^r \end{bmatrix}$ be 3 given matrices. Then $\sum_{r=1}^{50} tr.((AB)^r C_r) = 3(49.3^{50} + 1)$.(B) Given $A = \begin{bmatrix} 2 & 1 \\ 2 & 1 \end{bmatrix}$; $B = \begin{bmatrix} 9 & 3 \\ 3 & 1 \end{bmatrix}$. $I$ is a unit matrix of order 2. If $AX = A$, then $X = \begin{bmatrix} a & b \\ 2-2a & 1-2b \end{bmatrix}$ for $a, b \in R$(C) Given $A = \begin{bmatrix} 2 & 1 \\ 2 & 1 \end{bmatrix}$; $B = \begin{bmatrix} 9 & 3 \\ 3 & 1 \end{bmatrix}$. $I$ is a unit matrix of order 2. If $XA = I$, then $X$ does not exist(D) Given $A = \begin{bmatrix} 2 & 1 \\ 2 & 1 \end{bmatrix}$; $B = \begin{bmatrix} 9 & 3 \\ 3 & 1 \end{bmatrix}$. $I$ is a unit matrix of order 2. If $XB = 0$ but $BX \neq 0$, then $X = \begin{bmatrix} a & -3a \\ c & -3c \end{bmatrix}$, $a, c \in R, 3a + c \neq 0; 3b + d \neq 0$
Let $R = \begin{bmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{bmatrix}$ be a non-zero $3 \times 3$ matrix, where $x \sin \theta = y \sin \left( \theta + \frac{2\pi}{3} \right) = z \sin \left( \theta + \frac{4\pi}{3} \right) \neq 0, \theta \in (0, 2\pi)$. For a square matrix $M$, let trace $(M)$ denote the sum of all the diagonal elements of $M$. Then, among the statements: (I) Trace $(R) = 0$ (II) If trace $(\text{adj}(\text{adj}(R))) = 0$, then $R$ has exactly one non-zero entry.