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 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) $10$

Let $a , b$ be real numbers, and the system of equations $\left\{ \begin{array} { l } a x + 5 y + 12 z = 4 \\ x + a y + \frac { 8 } { 3 } z = 7 \\ 3 x + 8 y + a z = 1 \end{array} \right.$ has exactly one solution. After a series of Gaussian elimination operations, the original augmented matrix can be transformed to $\left[ \begin{array} { c c c | c } 1 & 2 & b & 7 \\ 0 & b & 5 & - 5 \\ 0 & 0 & b & 0 \end{array} \right]$ . Then $a = ( 14 - 1 ) , b = \frac { ( 14 - 2 ) } { ( 14 - 3 ) }$ . (Express as a fraction in lowest terms)

Consider a real $2 \times 2$ matrix $\left[ \begin{array} { l l } a & b \\ c & d \end{array} \right]$. If $\left[ \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right] \left[ \begin{array} { l l } a & b \\ c & d \end{array} \right] \left[ \begin{array} { c c } 1 & 0 \\ 0 & - 2 \end{array} \right] = \left[ \begin{array} { c c } 3 & - 4 \\ - 9 & - 7 \end{array} \right]$, what is the value of $c - 2b$?
(1) $- 11$ (2) $- 4$ (3) $1$ (4) $10$ (5) $11$

Let $a , b , c , d , r , s , t$ all be real numbers. It is known that three non-zero vectors $\vec { u } = ( a , b , 0 )$, $\vec { v } = ( c , d , 0 )$, and $\vec { w } = ( r , s , t )$ in coordinate space satisfy the dot products $\vec { w } \cdot \vec { u } = \vec { w } \cdot \vec { v } = 0$. Consider the $3 \times 3$ matrix $A = \left[ \begin{array} { l l l } a & b & 0 \\ c & d & 0 \\ r & s & t \end{array} \right]$. Select the correct options.
(1) If $\vec { u } \cdot \vec { v } = 0$, then the determinant $\left| \begin{array} { l l } a & b \\ c & d \end{array} \right| \neq 0$
(2) If $t \neq 0$, then the determinant $\left| \begin{array} { l l } a & b \\ c & d \end{array} \right| \neq 0$
(3) If there exists a vector $\overrightarrow { w ^ { \prime } }$ satisfying $\overrightarrow { w ^ { \prime } } \cdot \vec { u } = \overrightarrow { w ^ { \prime } } \cdot \vec { v } = 0$ and cross product $\overrightarrow { w ^ { \prime } } \times \vec { w } \neq \overrightarrow { 0 }$, then the determinant $\left| \begin{array} { l l } a & b \\ c & d \end{array} \right| \neq 0$
(4) If for any three real numbers $e , f , g$, the vector $( e , f , g )$ can be expressed as a linear combination of $\vec { u } , \vec { v } , \vec { w }$, then the determinant $\left| \begin{array} { l l } a & b \\ c & d \end{array} \right| \neq 0$
(5) If the determinant $\left| \begin{array} { l l } a & b \\ c & d \end{array} \right| \neq 0$, then the determinant of $A$ is not equal to 0

On the coordinate plane, let $A$ and $B$ denote the rotation matrices for clockwise and counterclockwise rotation by $90^{\circ}$ about the origin respectively. Let $C$ and $D$ denote the reflection matrices with reflection axes $x = y$ and $x = -y$ respectively. Select the correct options.
(1) $A$ and $C$ map the point $(1,0)$ to the same point
(2) $A = -B$
(3) $C = D^{-1}$
(4) $AB = CD$
(5) $AC = BD$

Given that $a , b , c , d$ are real numbers, and $\left[ \begin{array} { l l } 1 & - 1 \\ 3 & - 2 \end{array} \right] \left[ \begin{array} { l } a \\ b \end{array} \right] = \left[ \begin{array} { l } 1 \\ 0 \end{array} \right]$. If $\left[ \begin{array} { l l } 1 & - 1 \\ 3 & - 2 \end{array} \right] \left[ \begin{array} { l } 2 a + 1 \\ 2 b + 1 \end{array} \right] = \left[ \begin{array} { l } c \\ d \end{array} \right]$, then the value of $c - 3 d$ is (13-1)(13-2).

Let $A$ be a $3 \times 2$ matrix such that $A \left[ \begin{array} { c c } 1 & 0 \\ - 1 & 1 \end{array} \right] = \left[ \begin{array} { c c } 4 & - 6 \\ - 2 & 1 \\ 3 & 5 \end{array} \right]$ . If $A \left[ \begin{array} { l } 1 \\ 0 \end{array} \right] = \left[ \begin{array} { l } a \\ b \\ c \end{array} \right]$ , what is the value of $a + b + c$?
(1) 0
(2) 2
(3) 4
(4) 5
(5) 8

Let the second-order matrices $A = \left[ \begin{array} { l l } 1 & 0 \\ 1 & 0 \end{array} \right], B = \left[ \begin{array} { l l } 0 & 1 \\ 0 & 1 \end{array} \right]$. Select the correct options.
(1) $A ^ { 2 } = A$
(2) $A + B = B + A$
(3) $A B = B A$
(4) $( A - B ) ^ { 2 } = A ^ { 2 } - 2 A B + B ^ { 2 }$
(5) $( A + B ) ^ { 2 } = 2 ( A + B )$

Consider a discrete-time system where stochastic transitions between the two states (A and B) occur as shown in Figure 2.1. The transition probability in unit time from the state A to B is $\alpha$ and from the state B to A is $\beta$. Note that $0 < \alpha < 1$ and $0 < \beta < 1$. Variables $n$ and $k$ represent discrete time and are integers greater than or equal to 0.
Answer the following questions.

1. Let $P_{\mathrm{A}}(n)$ be the probability that the state is A at time $n$ and $P_{\mathrm{B}}(n)$ be the probability that the state is B at time $n$. Let $\boldsymbol{P}(n) = \binom{P_{\mathrm{A}}(n)}{P_{\mathrm{B}}(n)}$. Express matrix $\boldsymbol{M}$ using $\alpha$ and $\beta$, assuming $\boldsymbol{P}(n+1) = \boldsymbol{M}\boldsymbol{P}(n)$.
2. Obtain all eigenvalues and the corresponding eigenvectors of matrix $\boldsymbol{M}$.
3. As time tends towards infinity, the probability that the state is A and the probability that the state is B converge towards constant values. Obtain each value.
4. Assume $R_{\mathrm{A}}(n) = P_{\mathrm{A}}(n) - \lim_{k \rightarrow \infty} P_{\mathrm{A}}(k)$. Express $R_{\mathrm{A}}(n+1)$ by using $R_{\mathrm{A}}(n)$.

Assume vectors $\boldsymbol{a}_1, \boldsymbol{a}_2, \ldots, \boldsymbol{a}_m$ are linearly independent in a vector space $V$, where $m$ is an integer greater than or equal to 3. Obtain the condition that $m$ must satisfy in order for $\boldsymbol{a}_1 + \boldsymbol{a}_2,\ \boldsymbol{a}_2 + \boldsymbol{a}_3,\ \ldots,\ \boldsymbol{a}_{m-1} + \boldsymbol{a}_m$ and $\boldsymbol{a}_m + \boldsymbol{a}_1$ to be linearly independent.

Answer the following questions.
(1) The function $f ( x , y )$ with real variables $x , y$ is defined as follows:
$$f ( x , y ) = \left| \begin{array} { c c c } 1 & x _ { 1 } & y _ { 1 } \\ 1 & x _ { 2 } & y _ { 2 } \\ 1 & x & y \end{array} \right|$$
Show that the set of solutions of the equation $f ( x , y ) = 0$ is a line passing through two points $\left( x _ { 1 } , y _ { 1 } \right) , \left( x _ { 2 } , y _ { 2 } \right)$ on the $x y$ plane, where $x _ { 1 } \neq x _ { 2 }$.
(2) Find the value of the determinant $\left| \begin{array} { c c c } 1 & x _ { 1 } & x _ { 1 } ^ { 2 } \\ 1 & x _ { 2 } & x _ { 2 } ^ { 2 } \\ 1 & x _ { 3 } & x _ { 3 } ^ { 2 } \end{array} \right|$ in factored form.
(3) Show that there is a unique curve $y = a _ { 0 } + a _ { 1 } x + a _ { 2 } x ^ { 2 }$ passing through three points $\left( x _ { 1 } , y _ { 1 } \right) , \left( x _ { 2 } , y _ { 2 } \right) , \left( x _ { 3 } , y _ { 3 } \right)$ on the $x y$ plane, where $a _ { 0 } , a _ { 1 } , a _ { 2 }$ are constants and $x _ { 1 } , x _ { 2 } , x _ { 3 }$ are all distinct.
(4) The curve in (3) can be represented in the form $y = c _ { 1 } y _ { 1 } + c _ { 2 } y _ { 2 } + c _ { 3 } y _ { 3 }$, where each of $c _ { 1 } , c _ { 2 } , c _ { 3 }$ does not depend on $y _ { 1 } , y _ { 2 } , y _ { 3 }$. Find $c _ { 1 } , c _ { 2 } , c _ { 3 }$.
(5) Let us represent a curve $y = a _ { 0 } + a _ { 1 } x + a _ { 2 } x ^ { 2 } + a _ { 3 } x ^ { 3 } + a _ { 4 } x ^ { 4 }$ passing through five points $\left( x _ { 1 } , y _ { 1 } \right) , \ldots , \left( x _ { 5 } , y _ { 5 } \right)$ on the $x y$ plane in the form $y = c _ { 1 } y _ { 1 } + \cdots + c _ { 5 } y _ { 5 }$, where each of $c _ { 1 } , \ldots , c _ { 5 }$ does not depend on $y _ { 1 } , \ldots , y _ { 5 }$, and $x _ { 1 } , \ldots , x _ { 5 }$ are all distinct. Find $c _ { 1 }$.

Let $\mathbb { R } ^ { 3 }$ be the set of the three-dimensional real column vectors and $\mathbb { R } ^ { 3 \times 3 }$ be the set of the three-by-three real matrices. Let $n _ { 1 } , n _ { 2 }$, and $n _ { 3 } \in \mathbb { R } ^ { 3 }$ be linearly independent unit-length vectors and $n _ { 4 } \in \mathbb { R } ^ { 3 }$ be a unit-length vector not parallel to $n _ { 1 } , n _ { 2 }$, or $n _ { 3 }$. Let A and B be square matrices defined as
$$\mathbf { A } = \left( \begin{array} { l } n _ { 1 } ^ { \mathrm { T } } - n _ { 2 } ^ { \mathrm { T } } \\ n _ { 2 } ^ { \mathrm { T } } - n _ { 3 } ^ { \mathrm { T } } \\ n _ { 3 } ^ { \mathrm { T } } - n _ { 4 } ^ { \mathrm { T } } \end{array} \right) , \quad \mathbf { B } = \sum _ { i = 1 } ^ { 4 } n _ { i } n _ { i } ^ { \mathrm { T } }$$
Here, $\mathrm { X } ^ { \mathrm { T } }$ and $\boldsymbol { x } ^ { \mathrm { T } }$ denote the transpose of a matrix X and a vector $\boldsymbol { x }$, respectively. Answer the following questions.
(1) Find the condition for $n _ { 4 }$ such that the rank of $\mathbf { A }$ is three.
(2) In the three-dimensional Euclidean space $\mathbb { R } ^ { 3 }$, consider four planes $\Pi _ { i } = \{ x \in \left. \mathbb { R } ^ { 3 } \mid n _ { i } ^ { \mathrm { T } } \boldsymbol { x } - d _ { i } = 0 \right\}$ ( $d _ { i }$ is a real number, and $i = 1,2,3,4$ ) that satisfy the following three conditions: (i) the rank of A is three, (ii) $\Omega = \left\{ x \in \mathbb { R } ^ { 3 } \mid n _ { i } ^ { \mathrm { T } } x - d _ { i } \geq 0 , i = 1,2,3,4 \right\}$ is not the empty set, and (iii) there exists a sphere $\mathrm { C } ( \subset \Omega )$ to which $\Pi _ { i } ( i = 1,2,3,4 )$ are tangent. The position vector of the center of C is represented by $\mathbf { A } ^ { -1 } \boldsymbol { u }$ using a vector $\boldsymbol { u } \in \mathbb { R } ^ { 3 }$. Express $\boldsymbol { u }$ using $d _ { i } ( i = 1,2,3,4 )$.
(3) Show that B is a positive definite symmetric matrix.
(4) Consider the point P from which the sum of squared distances to four planes $\{ x \in \left. \mathbb { R } ^ { 3 } \mid n _ { i } ^ { \mathrm { T } } x - d _ { i } = 0 \right\}$ ( $d _ { i }$ is a real number, and $i = 1,2,3,4$ ) is minimized. The position vector of P is represented by $\mathrm { B } ^ { -1 } v$ using a vector $v \in \mathbb { R } ^ { 3 }$. Express $v$ using $n _ { i }$ and $d _ { i } ( i = 1,2,3,4 )$.
(5) Let $l _ { i }$ be a straight line through a point $Q _ { i }$, the position vector of which is $x _ { i } \in \mathbb { R } ^ { 3 }$, parallel to $n _ { i } ( i = 1,2,3 )$ in $\mathbb { R } ^ { 3 }$. Let $\mathrm { R } _ { i }$ be the orthogonal projection of an arbitrary point $R$, the position vector of which is $y \in \mathbb { R } ^ { 3 }$, onto $l _ { i }$. The position vector of $R _ { i }$ is represented by $y - \mathrm { W } _ { i } \left( y - x _ { i } \right)$ using a matrix $\mathrm { W } _ { i } \in \mathbb { R } ^ { 3 \times 3 }$. The identity matrix is denoted by $I \in \mathbb { R } ^ { 3 \times 3 }$.
(a) Express $\mathrm { W } _ { i }$ using $n _ { i }$ and I.
(b) Show that $\mathrm { W } _ { i } ^ { \mathrm { T } } \mathrm { W } _ { i } = \mathrm { W } _ { i }$.
(c) Consider a plane $\Sigma = \left\{ \boldsymbol { x } \in \mathbb { R } ^ { 3 } \mid \boldsymbol { a } ^ { \mathrm { T } } \boldsymbol { x } = b \right\} \left( \boldsymbol { a } \in \mathbb { R } ^ { 3 } \right.$ is a non-zero vector, and $b$ is a real number). Let $\mathrm { S } \in \Sigma$ be the point from which the sum of squared distances to $l _ { 1 } , l _ { 2 }$, and $l _ { 3 }$ is minimized. When $n _ { 1 } , n _ { 2 }$, and $n _ { 3 }$ are orthogonal to each other, the position vector of $S$ is represented by
$$\left( \mathrm { I } - \frac { a a ^ { \mathrm { T } } } { a ^ { \mathrm { T } } a } \right) w + \frac { a b } { a ^ { \mathrm { T } } a }$$
using a vector $\boldsymbol { w } \in \mathbb { R } ^ { 3 }$ which is independent of $a$ and $b$. Express $\boldsymbol { w }$ using $\mathbf { W } _ { i }$ and $x _ { i } ( i = 1,2,3 )$.

A line on a two-dimensional plane can be expressed as $\alpha x + \beta y + \gamma = 0$, where $( x , y )$ is a point on the line in the Cartesian coordinate system. We call the column vector $( \alpha , \beta , \gamma ) ^ { \mathrm { T } }$ a coefficient vector of the line. Answer the following questions. Note that the coefficient vector in your answer must satisfy $\alpha ^ { 2 } + \beta ^ { 2 } = 1$.
(1) Find a coefficient vector of a line that passes through a point $\vec { a }$ and is perpendicular to a unit vector $\vec { v }$ on a two-dimensional plane.
(2) Let a line B pass through a point $\vec { b }$ and be perpendicular to a unit vector $\vec { n }$. Given a line $A$, let the line $A ^ { \prime }$ be the mirror transformation of the line $A$ over the line $B$. Using $\vec { b }$ and $\vec { n }$, write a three-dimensional square matrix that transforms a coefficient vector of the line A to a coefficient vector of the line $\mathrm { A } ^ { \prime }$.
(3) Find the determinant of the matrix derived in Question (2).
(4) Consider the movement of the line $\mathrm { D } _ { t }$ whose coefficient vector changes with the real variable $t$ as $\left( 4 t , 4 t ^ { 2 } - 1 , t \right) ^ { \mathrm { T } }$. This line passes through a point regardless of $t$. Find the coordinate of that point.
(5) Suppose that, with the mirror transformation over a line $M _ { t }$, which also changes with $t$, the line $\mathrm { D } _ { t }$ in Question (4) is transformed to the line with a coefficient vector $( 0,1 , - t ) ^ { \mathrm { T } }$. Find the coefficient vector $\left( \alpha _ { t } , \beta _ { t } , \gamma _ { t } \right) ^ { \mathrm { T } }$ of the line $\mathrm { M } _ { t }$, where $\alpha _ { t } > 0$ and $\beta _ { t } > 0$ for $t > 0$.
(6) When $t$ changes from 0 to $+ \infty$, consider the region where the line $\mathrm { M } _ { t }$ in Question (5) can exist. Describe the region using a simple mathematical expression and draw a diagram of the region.

$$A = \begin{bmatrix} 2 & 4 \\ 1 & 3 \end{bmatrix}$$
Given that $A^{t}$ is the transpose of the matrix and $A^{-1}$ is its inverse matrix, which of the following is the product $A^{t} \cdot A^{-1}$?
A) $\begin{bmatrix} \frac{5}{2} & -3 \\ \frac{9}{2} & -5 \end{bmatrix}$
B) $\begin{bmatrix} \frac{3}{2} & -2 \\ 1 & 3 \end{bmatrix}$
C) $\begin{bmatrix} -2 & \frac{-9}{2} \\ 3 & \frac{5}{2} \end{bmatrix}$
D) $\begin{bmatrix} \frac{9}{2} & 3 \\ \frac{-5}{2} & -1 \end{bmatrix}$
E) $\begin{bmatrix} -3 & -1 \\ \frac{5}{2} & -2 \end{bmatrix}$

$$A = \left[ \begin{array} { l l } 1 & 1 \\ 0 & 1 \end{array} \right], \quad B = \left[ \begin{array} { l l } 1 & 0 \\ \cdots & \cdots \end{array} \right]$$

Let a, b and c be positive real numbers,
$$\left[ \begin{array} { l l } a & b \\ 0 & c \end{array} \right] \cdot \left[ \begin{array} { l l } a & b \\ 0 & c \end{array} \right] = \left[ \begin{array} { l l } 1 & 2 \\ 0 & 4 \end{array} \right]$$
The matrix equation is given. Accordingly, what is the sum $a + b + c$?
A) $\frac { 11 } { 3 }$
B) $\frac { 7 } { 4 }$
C) 4
D) 5
E) 6

For a matrix A with multiplicative inverse $A^{-1}$,
$$\left[ \begin{array} { l l } 2 & 1 \end{array} \right] \cdot \left[ \begin{array} { l l } 1 & 0 \\ 3 & 1 \end{array} \right] ^ { - 1 } \cdot \left[ \begin{array} { l } 1 \\ 4 \end{array} \right] = [ a ]$$
In the matrix equation, what is a?
A) 1
B) 2
C) 3
D) 4
E) 5

$$\begin{aligned} & A = \left[ \begin{array} { l l } 2 & 3 \\ 1 & 2 \end{array} \right] \\ & B = \left[ \begin{array} { l l } 1 & 2 \\ 0 & 5 \end{array} \right] \end{aligned}$$
With the matrix notation
$$( 2 A - B ) \cdot \left[ \begin{array} { l } x \\ y \end{array} \right] = \left[ \begin{array} { l } 1 \\ 0 \end{array} \right]$$
Which of the following is the system of linear equations?
A) $\begin{aligned} & x - 4 y = 0 \\ & 2 x - y = 1 \end{aligned}$
B) $\begin{aligned} & x + 2 y = 0 \\ & 2 x - 3 y = 1 \end{aligned}$
C) $\begin{aligned} & 2 x + y = 1 \\ & x - y = 0 \end{aligned}$
D) $\begin{aligned} & 3 x - 2 y = 1 \\ & 2 x + y = 0 \end{aligned}$
E) $\begin{aligned} & 3 x + 4 y = 1 \\ & 2 x - y = 0 \end{aligned}$

$$A = \left[ \begin{array} { l l } 3 & 2 \\ 0 & 1 \end{array} \right]$$
Given this, what is the value of the determinant $\left| A - A ^ { \top } \right|$?
A) 3
B) 4
C) 5
D) 6
E) 9

Let A be a $3 \times 3$ matrix. Given that
$$\begin{aligned} & { \left[ \begin{array} { l l l } 2 & 1 & 3 \end{array} \right] \cdot A = \left[ \begin{array} { l l l } 0 & 2 & 2 \end{array} \right] } \\ & { \left[ \begin{array} { l l l } 1 & 4 & 0 \end{array} \right] \cdot A = \left[ \begin{array} { l l l } 3 & 1 & 5 \end{array} \right] } \end{aligned}$$
What is the product $\left[ \begin{array} { l l l } 5 & 6 & 6 \end{array} \right] \cdot A$ equal to?
A) $\left[ \begin{array} { l l l } 2 & 1 & 3 \end{array} \right]$
B) $\left[ \begin{array} { l l l } 3 & 3 & 7 \end{array} \right]$
C) $\left[ \begin{array} { l l l } 3 & 5 & 9 \end{array} \right]$
D) $\left[ \begin{array} { l l l } 6 & 2 & 10 \end{array} \right]$
E) $\left[ \begin{array} { l l l } 6 & 4 & 12 \end{array} \right]$

Let I be the $2 \times 2$ identity matrix and
$$A = \left[ \begin{array} { l l } 4 & 5 \\ 1 & 3 \end{array} \right]$$
Accordingly, which of the following is $( \mathbf { A } - \mathbf { I } ) ^ { - \mathbf { 1 } }$ equal to?
A) $\left[ \begin{array} { r r } 2 & - 5 \\ - 1 & 3 \end{array} \right]$
B) $\left[ \begin{array} { r r } 1 & - 4 \\ - 2 & 3 \end{array} \right]$
C) $\left[ \begin{array} { r r } 0 & - 1 \\ - 1 & 4 \end{array} \right]$
D) $\left[ \begin{array} { l l } - 2 & 5 \\ - 1 & 0 \end{array} \right]$
E) $\left[ \begin{array} { l l } 2 & - 5 \\ 0 & - 3 \end{array} \right]$

$$\left[ \begin{array} { l l } 3 & 2 \\ 1 & 0 \end{array} \right] \cdot \mathrm { A } = \left[ \begin{array} { c c } - 2 & 4 \\ 1 & 5 \end{array} \right]$$
What is the determinant of matrix A that satisfies this equation?
A) 4
B) 5
C) 6
D) 7
E) 8

Let A be a $2 \times 2$ matrix and $I$ be the $2 \times 2$ identity matrix such that
$$A ^ { 2 } = \left[ \begin{array} { l l } 2 & 1 \\ 1 & 5 \end{array} \right]$$
What is the value of the determinant $| ( \mathbf { A } - \mathbf { I } ) ( \mathbf { A } + \mathbf { I } ) |$?
A) 2
B) 3
C) 4
D) 5
E) 6

Let $A$ and $B$ be $2 \times 1$ matrices and $t$ be a variable such that for all $x$ and $y$ values satisfying
$$x - y = 3$$
we have
$$\left[ \begin{array} { l } x \\ y \end{array} \right] = t A + B$$
Accordingly, which of the following could matrices A and B be, respectively?
A) $\left[ \begin{array} { l } 1 \\ 0 \end{array} \right] , \left[ \begin{array} { l } 3 \\ 0 \end{array} \right]$
B) $\left[ \begin{array} { l } 0 \\ 1 \end{array} \right] , \left[ \begin{array} { l } 3 \\ 0 \end{array} \right]$
C) $\left[ \begin{array} { l } 1 \\ 1 \end{array} \right] , \left[ \begin{array} { l } 3 \\ 1 \end{array} \right]$
D) $\left[ \begin{array} { l } 1 \\ 1 \end{array} \right] , \left[ \begin{array} { l } 3 \\ 0 \end{array} \right]$
E) $\left[ \begin{array} { l } 1 \\ 1 \end{array} \right] , \left[ \begin{array} { r } 3 \\ - 3 \end{array} \right]$