Let $A = \left[ \begin{array} { c c } i & - i \\ - i & i \end{array} \right] , i = \sqrt { - 1 }$. Then, the system of linear equations $A ^ { 8 } \left[ \begin{array} { l } x \\ y \end{array} \right] = \left[ \begin{array} { c } 8 \\ 64 \end{array} \right]$ has :
(1) A unique solution
(2) Infinitely many solutions
(3) No solution
(4) Exactly two solutions

Let $A = \begin{pmatrix} 0 & -2 \\ 2 & 0 \end{pmatrix}$. If $M$ and $N$ are two matrices given by $M = \sum _ { k = 1 } ^ { 10 } A ^ { 2k }$ and $N = \sum _ { k = 1 } ^ { 10 } A ^ { 2k - 1 }$ then $MN^{2}$ is
(1) a non-identity symmetric matrix
(2) a skew-symmetric matrix
(3) neither symmetric nor skew-symmetric matrix
(4) an identity matrix

Let $A = \left[ a _ { i j } \right]$ be a square matrix of order 3 such that $a _ { i j } = 2 ^ { j - i }$, for all $i , j = 1,2,3$. Then, the matrix $A ^ { 2 } + A ^ { 3 } + \ldots + A ^ { 10 }$ is equal to
(1) $\left( \frac { 3 ^ { 10 } - 1 } { 2 } \right) A$
(2) $\left( \frac { 3 ^ { 10 } + 1 } { 2 } \right) A$
(3) $\left( \frac { 3 ^ { 10 } + 3 } { 2 } \right) A$
(4) $\left( \frac { 3 ^ { 10 } - 3 } { 2 } \right) A$

Let the matrix $A = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}$ and the matrix $B_0 = A^{49} + 2A^{98}$. If $B_n = \text{Adj}(B_{n-1})$ for all $n \geq 1$, then $\det(B_4)$ is equal to
(1) $3^{28}$
(2) $3^{30}$
(3) $3^{32}$
(4) $3^{36}$

Let $A = \begin{pmatrix} 2 & -1 & -1 \\ 1 & 0 & -1 \\ 1 & -1 & 0 \end{pmatrix}$ and $B = A - I$. If $\omega = \frac { \sqrt { 3 }\, i - 1 } { 2 }$, then the number of elements in the set $\left\{ n \in \{1,2,\ldots,100\} : A ^ { n } + \omega B ^ { n } = A + B \right\}$ is equal to $\_\_\_\_$.

Let $P$ be a square matrix such that $P^{2} = I - P$. For $\alpha, \beta, \gamma, \delta \in \mathbb{N}$, if $P^{\alpha} + P^{\beta} = \gamma I - 29P$ and $P^{\alpha} - P^{\beta} = \delta I - 13P$, then $\alpha + \beta + \gamma - \delta$ is equal to
(1) 18
(2) 40
(3) 22
(4) 24

Let $A = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$. If $M = \sum_{k=1}^{20} (A^k + B^k)$, then $\det(M)$ is equal to
(1) 100
(2) 200
(3) 0
(4) 400

Let $\mathrm { A } = \left[ \begin{array} { c c } \frac { 1 } { \sqrt { 10 } } & \frac { 3 } { \sqrt { 10 } } \\ \frac { - 3 } { \sqrt { 10 } } & \frac { 1 } { \sqrt { 10 } } \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { c c } 1 & - \mathrm { i } \\ 0 & 1 \end{array} \right]$, where $\mathrm { i } = \sqrt { - 1 }$. If $\mathrm { M } = \mathrm { A } ^ { \mathrm { T } } \mathrm { BA }$, then the inverse of the matrix $\mathrm { AM } ^ { 2023 } \mathrm {~A} ^ { \mathrm { T } }$ is
(1) $\left[ \begin{array} { c c } 1 & - 2023 i \\ 0 & 1 \end{array} \right]$
(2) $\left[ \begin{array} { l l } 1 & 0 \\ - 2023 i & 1 \end{array} \right]$
(3) $\left[ \begin{array} { l l } 1 & 0 \\ 2023 i & 1 \end{array} \right]$
(4) $\left[ \begin{array} { c c } 1 & 2023 i \\ 0 & 1 \end{array} \right]$

Let $A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 4 & -1 \\ 0 & 12 & -3 \end{pmatrix}$. Then the sum of the diagonal elements of the matrix $(A+I)^{11}$ is equal to:
(1) 6144
(2) 4094
(3) 4097
(4) 2050

Let $P = \left[ \begin{array} { c c } \frac { \sqrt { 3 } } { 2 } & \frac { 1 } { 2 } \\ - \frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 } \end{array} \right] , A = \left[ \begin{array} { l l } 1 & 1 \\ 0 & 1 \end{array} \right]$ and $Q = P A P ^ { T }$. If $P ^ { T } Q ^ { 2007 } P = \left[ \begin{array} { l l } a & b \\ c & d \end{array} \right]$ then $2 a + b - 3 c - 4 d$ is equal to
(1) 2004
(2) 2005
(3) 2007
(4) 2006

Let $A = \left[ \begin{array} { c c } 1 & \frac { 1 } { 51 } \\ 0 & 1 \end{array} \right]$. If $B = \left[ \begin{array} { c c } 1 & 2 \\ - 1 & - 1 \end{array} \right] A \left[ \begin{array} { c c } - 1 & - 2 \\ 1 & 1 \end{array} \right]$, then the sum of all the elements of the matrix $\sum _ { n = 1 } ^ { 50 } B ^ { n }$ is equal to
(1) 75
(2) 125
(3) 50
(4) 100

Let $A = \left[ \begin{array} { l l } 1 & 2 \\ 0 & 1 \end{array} \right]$ and $B = I + \operatorname { adj } ( A ) + ( \operatorname { adj } A ) ^ { 2 } + \ldots + ( \operatorname { adj } A ) ^ { 10 }$. Then, the sum of all the elements of the matrix $B$ is:
(1) - 124
(2) 22
(3) - 88
(4) - 110

Let $A = \left[ \begin{array} { c c } 2 & - 1 \\ 1 & 1 \end{array} \right]$. If the sum of the diagonal elements of $A ^ { 13 }$ is $3 ^ { n }$, then $n$ is equal to $\_\_\_\_$

Let $A = \left[ \begin{array} { c c } \frac { 1 } { \sqrt { 2 } } & - 2 \\ 0 & 1 \end{array} \right]$ and $P = \left[ \begin{array} { c c } \cos \theta & - \sin \theta \\ \sin \theta & \cos \theta \end{array} \right] , \theta > 0$. If $\mathrm { B } = \mathrm { PAP } ^ { \mathrm { T } } , \mathrm { C } = \mathrm { P } ^ { \mathrm { T } } \mathrm { B } ^ { 10 } \mathrm { P }$ and the sum of the diagonal elements of $C$ is $\frac { \mathrm { m } } { \mathrm { n } }$, where $\operatorname { gcd } ( \mathrm { m } , \mathrm { n } ) = 1$, then $\mathrm { m } + \mathrm { n }$ is :
(1) 127
(2) 258
(3) 65
(4) 2049

Let $A = \left[ a _ { i j } \right]$ be a matrix of order $3 \times 3$, with $a _ { i j } = ( \sqrt { 2 } ) ^ { i + j }$. If the sum of all the elements in the third row of $A ^ { 2 }$ is $\alpha + \beta \sqrt { 2 } , \alpha , \beta \in \mathbf { Z }$, then $\alpha + \beta$ is equal to :
(1) 280
(2) 224
(3) 210
(4) 168

Let $\mathrm{S} = \{\mathrm{m} \in \mathbf{Z}: \mathrm{A}^{\mathrm{m}^2} + \mathrm{A}^{\mathrm{m}} = 3\mathrm{I} - \mathrm{A}^{-6}\}$, where $\mathrm{A} = \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix}$. Then $\mathrm{n}(\mathrm{S})$ is equal to \_\_\_\_ .

Which of the following matrices is equal to $\left[ \begin{array} { c c } - 1 & 0 \\ 1 & - 1 \end{array} \right] ^ { 5 }$ ?
(1) $\left[ \begin{array} { c c } - 1 & 0 \\ - 5 & - 1 \end{array} \right]$
(2) $\left[ \begin{array} { c c } 1 & 0 \\ - 5 & 1 \end{array} \right]$
(3) $\left[ \begin{array} { c c } - 1 & 5 \\ 0 & - 1 \end{array} \right]$
(4) $\left[ \begin{array} { l l } 1 & 0 \\ 5 & 1 \end{array} \right]$
(5) $\left[ \begin{array} { c c } - 1 & 0 \\ 5 & - 1 \end{array} \right]$

Let matrix $A = \left[ \begin{array} { c c } 1 & - 2 \\ 0 & 1 \end{array} \right] \left[ \begin{array} { l l } 1 & 0 \\ 0 & 6 \end{array} \right] \left[ \begin{array} { c c } 1 & - 2 \\ 0 & 1 \end{array} \right] ^ { - 1 } , B = \left[ \begin{array} { c c } 1 & - 2 \\ 0 & 1 \end{array} \right] \left[ \begin{array} { l l } 6 & 0 \\ 0 & 1 \end{array} \right] \left[ \begin{array} { c c } 1 & - 2 \\ 0 & 1 \end{array} \right] ^ { - 1 }$, where $\left[ \begin{array} { c c } 1 & - 2 \\ 0 & 1 \end{array} \right] ^ { - 1 }$ is the inverse matrix of $\left[ \begin{array} { c c } 1 & - 2 \\ 0 & 1 \end{array} \right]$. If $A + B = \left[ \begin{array} { l l } a & b \\ c & d \end{array} \right]$, then $a + b + c + d =$ (10)(11).

An electronics company has several hundred employees with two types of meal arrangements: bringing meals from home or eating out. Long-term surveys have found that: if an employee brings meals from home on a given day, then $10\%$ will switch to eating out the next day; if an employee eats out on a given day, then $20\%$ will switch to bringing meals from home the next day. Let $x _ { 0 }$、$y _ { 0 }$ respectively represent the proportion of employees bringing meals from home and eating out today relative to the total number of employees, where $x _ { 0 }$、$y _ { 0 }$ are both positive, and $x _ { n }$、$y _ { n }$ respectively represent the proportion of employees bringing meals from home and eating out after $n$ days relative to the total number of employees. Given that the number of employees in the company remains unchanged, select the correct options.
(1) $y _ { 1 } = 0.9 y _ { 0 } + 0.2 x _ { 0 }$
(2) $\left[ \begin{array} { l } x _ { n + 1 } \\ y _ { n + 1 } \end{array} \right] = \left[ \begin{array} { l l } 0.9 & 0.2 \\ 0.1 & 0.8 \end{array} \right] \left[ \begin{array} { l } x _ { n } \\ y _ { n } \end{array} \right]$
(3) If $\frac { x _ { 0 } } { y _ { 0 } } = \frac { 2 } { 1 }$ , then $\frac { x _ { n } } { y _ { n } } = \frac { 2 } { 1 }$ holds for any positive integer $n$
(4) If $y _ { 0 } > x _ { 0 }$ , then $y _ { 1 } > x _ { 1 }$
(5) If $x _ { 0 } > y _ { 0 }$ , then $x _ { 0 } > x _ { 1 }$

Let $A = \left[ \begin{array} { l l } 1 & 2 \\ 0 & 3 \end{array} \right]$. If $A ^ { 4 } = \left[ \begin{array} { l l } a & b \\ c & d \end{array} \right]$, what is the value of $a + b + c + d$?
(1) 158
(2) 162
(3) 166
(4) 170
(5) 174

Let matrix $A = \left[\begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array}\right]$. If $A^{7} - 3A = \left[\begin{array}{ll} a & b \\ c & d \end{array}\right]$, then which of the following is the value of $a + b + c + d$?
(1) $-8$
(2) $-5$
(3) $5$
(4) $8$
(5) $10$