Question:

Which of the following options is/are correct ?<br>(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 $\sum_{r=1}^{50} tr.((AB)^r C_r) = 3(49.3^{50} + 1)$.<br>(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 $a, b \in R$<br>(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<br>(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 $X = \begin{bmatrix} a & -3a \\ c & -3c \end{bmatrix}$, $a, c \in R, 3a + c \neq 0; 3b + d \neq 0$