Let $A = \left[ \begin{array} { l l l } 2 & a & 0 \\ 1 & 3 & 1 \\ 0 & 5 & b \end{array} \right]$. If $A ^ { 3 } = 4 A ^ { 2 } - A - 21 I$, where $I$ is the identity matrix of order $3 \times 3$, then $2 a + 3 b$ is equal to
(1) $- 9$
(2) $- 13$
(3) $- 10$
(4) $- 12$

Let $\alpha \beta \neq 0$ and $A = \left[ \begin{array} { r r r } \beta & \alpha & 3 \\ \alpha & \alpha & \beta \\ - \beta & \alpha & 2 \alpha \end{array} \right]$. If $B = \left[ \begin{array} { r r r } 3 \alpha & - 9 & 3 \alpha \\ - \alpha & 7 & - 2 \alpha \\ - 2 \alpha & 5 & - 2 \beta \end{array} \right]$ is the matrix of cofactors of the elements of $A$, then $\operatorname { det } ( A B )$ is equal to :
(1) 64
(2) 216
(3) 343
(4) 125

Let $\mathrm { A } = \left[ \begin{array} { c c c } 2 & 1 & 2 \\ 6 & 2 & 11 \\ 3 & 3 & 2 \end{array} \right]$ and $\mathrm { P } = \left[ \begin{array} { l l l } 1 & 2 & 0 \\ 5 & 0 & 2 \\ 7 & 1 & 5 \end{array} \right]$. The sum of the prime factors of $\left| \mathrm { P } ^ { - 1 } \mathrm { AP } - 2 \mathrm { I } \right|$ is equal to
(1) 26
(2) 27
(3) 66
(4) 23

$f ( x ) = \left| \begin{array} { c c c } 2 \cos ^ { 4 } x & 2 \sin ^ { 4 } x & 3 + \sin ^ { 2 } 2 x \\ 3 + 2 \cos ^ { 4 } x & 2 \sin ^ { 4 } x & \sin ^ { 2 } 2 x \\ 2 \cos ^ { 4 } x & 3 + 2 \sin ^ { 4 } x & \sin ^ { 2 } 2 x \end{array} \right|$ then $\frac { 1 } { 5 } f ^ { \prime } ( 0 )$ is equal to:
(1) 0
(2) 1
(3) 2
(4) 6

If the system of linear equations $$\begin{aligned} & x - 2 y + z = - 4 \\ & 2 x + \alpha y + 3 z = 5 \\ & 3 x - y + \beta z = 3 \end{aligned}$$ has infinitely many solutions, then $12 \alpha + 13 \beta$ is equal to
(1) 60
(2) 64
(3) 54
(4) 58

Let the system of equations $x + 2y + 3z = 5$, $2x + 3y + z = 9$, $4x + 3y + \lambda z = \mu$ have infinite number of solutions. Then $\lambda + 2\mu$ is equal to:
(1) 28
(2) 17
(3) 22
(4) 15

Consider the system of linear equation $x + y + z = 4 \mu , x + 2 y + 2 \lambda z = 10 \mu , x + 3 y + 4 \lambda ^ { 2 } z = \mu ^ { 2 } + 15$, where $\lambda , \mu \in \mathrm { R }$. Which one of the following statements is NOT correct?
(1) The system has unique solution if $\lambda \neq \frac { 1 } { 2 }$ and $\mu \neq 1$
(2) The system is inconsistent if $\lambda = \frac { 1 } { 2 }$ and $\mu \neq 1, 15$
(3) The system has infinite number of solutions if $\lambda = \frac { 1 } { 2 }$ and $\mu = 15$
(4) The system is consistent if $\lambda \neq \frac { 1 } { 2 }$

Let $A$ be a $3 \times 3$ real matrix such that $A\begin{pmatrix}0\\1\\0\end{pmatrix} = \begin{pmatrix}2\\0\\0\end{pmatrix}$, $A\begin{pmatrix}0\\0\\1\end{pmatrix} = \begin{pmatrix}4\\0\\0\end{pmatrix}$, $A\begin{pmatrix}1\\1\\1\end{pmatrix} = \begin{pmatrix}2\\1\\1\end{pmatrix}$. Then, the system $(A - 3I)\begin{pmatrix}x\\y\\z\end{pmatrix} = \begin{pmatrix}1\\2\\3\end{pmatrix}$ has
(1) unique solution
(2) exactly two solutions
(3) no solution
(4) infinitely many solutions

For $\alpha , \beta \in \mathbb { R }$ and a natural number $n$, let $A _ { r } = \left| \begin{array} { c c c } r & 1 & \frac { n ^ { 2 } } { 2 } + \alpha \\ 2 r & 2 & n ^ { 2 } - \beta \\ 3 r - 2 & 3 & \frac { n ( 3 n - 1 ) } { 2 } \end{array} \right|$. Then $\sum_{r=1}^{n} A_r$ is
(1) 0
(2) $4 \alpha + 2 \beta$
(3) $2 \alpha + 4 \beta$
(4) $2 n$

For a $3 \times 3$ matrix $M$, let trace ( $M$ ) denote the sum of all the diagonal elements of $M$. Let $A$ be a $3 \times 3$ matrix such that $| A | = \frac { 1 } { 2 }$ and trace $( A ) = 3$. If $B = \operatorname { adj } ( \operatorname { adj } ( 2 A ) )$, then the value of $| B | +$ trace $( \mathrm { B } )$ equals :
(1) 56
(2) 132
(3) 174
(4) 280

Let $A$ and $b$ be defined as
$$A = \left( \begin{array} { r r r } - 3 & 0 & 0 \\ - 2 & - 3 & 1 \\ 2 & - 3 & - 3 \end{array} \right) , b = \left( \begin{array} { l } 1 \\ 1 \\ 0 \end{array} \right) .$$
The partial derivative of a scalar-valued function $f ( x )$ with respect to $x = \left( \begin{array} { l l l } x _ { 1 } & x _ { 2 } & x _ { 3 } \end{array} \right) ^ { T }$ is defined as
$$\frac { \partial } { \partial x } f ( x ) = \left( \frac { \partial } { \partial x _ { 1 } } f ( x ) \quad \frac { \partial } { \partial x _ { 2 } } f ( x ) \quad \frac { \partial } { \partial x _ { 3 } } f ( x ) \right)$$
and a stationary point of $f ( x )$ is defined as $x$ satisfying $\frac { \partial } { \partial x } f ( x ) = \left( \begin{array} { l l l } 0 & 0 & 0 \end{array} \right) . x ^ { T }$ denotes the transpose of $x$. Answer the following questions.
(1) Find the characteristic polynomial of $A$.
(2) $C$ is given as $C = A ^ { 5 } + 9 A ^ { 4 } + 30 A ^ { 3 } + 36 A ^ { 2 } + 30 A + 9 I$ by using $A$ and an identity matrix $I$. Calculate $C$.
(3) Calculate the partial derivative of $x ^ { T } A x$ with respect to $x$.
(4) Find a symmetric matrix $\tilde { A }$ that satisfies equation $x ^ { T } A x = x ^ { T } \tilde { A } x$ for any vector $x$. Find eigenvalues $\lambda _ { 1 } , \lambda _ { 2 } , \lambda _ { 3 } \left( \lambda _ { 1 } \geq \lambda _ { 2 } \geq \lambda _ { 3 } \right)$, and eigenvectors $v _ { 1 } , v _ { 2 } , v _ { 3 }$. Choose the eigenvectors such that $V = \left( v _ { 1 } v _ { 2 } v _ { 3 } \right)$ becomes an orthogonal matrix.
(5) Prove that $x ^ { T } A x \leq 0$ holds for any real vector $x$.
(6) Find a stationary point of function $g ( x ) = x ^ { T } A x + 2 b ^ { T } x$.

Problem 2
Consider the column vectors $\mathbf { a } _ { 1 } = \left( \begin{array} { l } 0 \\ 1 \\ 1 \end{array} \right) , \mathbf { a } _ { 2 } = \left( \begin{array} { l } 1 \\ 0 \\ 1 \end{array} \right) , \mathbf { a } _ { 3 } = \left( \begin{array} { l } 1 \\ 1 \\ 0 \end{array} \right) , \mathbf { b } = \left( \begin{array} { l } 1 \\ 2 \\ 4 \end{array} \right) , \mathbf { 0 } = \left( \begin{array} { l } 0 \\ 0 \\ 0 \end{array} \right)$.
I. When $\mathbf { A } = \left( \begin{array} { l l l } \mathbf { a } _ { 1 } & \mathbf { a } _ { 2 } & \mathbf { a } _ { 3 } \end{array} \right)$, obtain the three-dimensional column vector $\mathbf { x }$ which meets
$$A x - b = 0 .$$
II. Any $m \times n$ real matrix $\mathbf { B }$ is expressed using orthonormal matrices $\mathbf { U } ( m \times m )$ and $\mathrm { V } ( n \times n )$ as
$$\mathbf { B } = \mathbf { U \Sigma V } ^ { T } , \quad \boldsymbol { \Sigma } = \left( \begin{array} { c c c c c c c } \sigma _ { 1 } & 0 & \cdots & 0 & 0 & \cdots & 0 \\ 0 & \sigma _ { 2 } & \ddots & \vdots & \vdots & & \vdots \\ \vdots & \ddots & \ddots & 0 & \vdots & & \vdots \\ 0 & \cdots & 0 & \sigma _ { r } & 0 & \cdots & 0 \\ 0 & \cdots & \cdots & 0 & 0 & \cdots & 0 \\ \vdots & & & \vdots & \vdots & & \vdots \\ 0 & \cdots & \cdots & 0 & 0 & \cdots & 0 \end{array} \right) , \quad r = \operatorname { rank } ( \mathbf { B } ) .$$
$\sigma _ { 1 } , \sigma _ { 2 } , \cdots , \sigma _ { r }$ are positive real numbers, and they are called singular values of $\mathbf { B }$. $\mathbf { P } ^ { T }$ means the transposed matrix of a matrix $\mathbf { P }$. Then, express $\mathbf { B B } ^ { T }$ and $\mathbf { B } ^ { T } \mathbf { B }$ using matrices $\mathbf { U } , \mathbf { V } , \boldsymbol { \Sigma }$ and their transposed matrices, respectively.
Let $\mathbf { B } = \left( \mathbf { a } _ { 1 } \mathbf { a } _ { 2 } \right)$ in the following questions.
III. Find the eigenvalues and corresponding eigenvectors for $\mathbf { B B } ^ { T }$.
IV. Find singular values of $\mathbf { B }$ and orthonormal matrices $\mathbf { U }$ and $\mathbf { V }$ used in Equation (2).
V. Find the two-dimensional column vector $\mathbf { x }$ which minimizes the norm
$$\| \mathrm { Bx } - \mathrm { b } \| ^ { 2 } = ( \mathrm { Bx } - \mathrm { b } ) ^ { T } ( \mathrm { Bx } - \mathrm { b } ) .$$

Problem 4
In a three-dimensional Cartesian coordinate system $x y z$, consider the positional relationship among three planes defined by Equations (1)-(3), and the positional relationship among the three planes and a sphere defined by Equation (4).
$$\begin{aligned} & a _ { 11 } x + a _ { 12 } y + a _ { 13 } z = b _ { 1 } , \\ & a _ { 21 } x + a _ { 22 } y + a _ { 23 } z = b _ { 2 } , \\ & a _ { 31 } x + a _ { 32 } y + a _ { 33 } z = b _ { 3 } , \\ & x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 3 , \end{aligned}$$
where $a _ { i j }$ and $b _ { i } ( i , j = 1,2,3 )$ are constants.
For the three planes, let $\mathrm { A } = \left( \begin{array} { l l l } a _ { 11 } & a _ { 12 } & a _ { 13 } \\ a _ { 21 } & a _ { 22 } & a _ { 23 } \\ a _ { 31 } & a _ { 32 } & a _ { 33 } \end{array} \right)$ be the coefficient matrix and $\mathbf { B } = \left( \begin{array} { l l l l } a _ { 11 } & a _ { 12 } & a _ { 13 } & b _ { 1 } \\ a _ { 21 } & a _ { 22 } & a _ { 23 } & b _ { 2 } \\ a _ { 31 } & a _ { 32 } & a _ { 33 } & b _ { 3 } \end{array} \right)$ be the augmented coefficient matrix.
I. Let $\mathbf { A } = \left( \begin{array} { l l l } 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{array} \right)$ and $\mathbf { B } = \left( \begin{array} { c c c c } 1 & 1 & 1 & 3 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & - c \end{array} \right)$ where $c$ is a positive constant.

1. Find $\operatorname { rank } ( \mathrm { A } )$ and $\operatorname { rank } ( \mathrm { B } )$.
2. Among the three planes, the plane that is tangential to the sphere defined by Equation (4) at a point $\mathrm { P } ( 1,1,1 )$ is called Plane 1. Between the other two planes, the plane with the shorter distance to P is called Plane 2. Find the distance between P and Plane 2. Then, find the volume of the part of the sphere existing between Planes 1 and 2.

II. When the three planes intersect in a line, find $\operatorname { rank } ( \mathrm { A } )$ and $\operatorname { rank } ( \mathrm { B } )$.
III. Suppose that the three planes are tangential to the sphere at three different points. Illustrate all possible positional relationships among the three planes and the sphere. In addition, for each relationship, find $\operatorname { rank } ( \mathrm { A } )$ and $\operatorname { rank } ( \mathrm { B } )$.

Consider to solve the following simultaneous linear equation:
$$A x = b$$
where $\boldsymbol { A } \in \mathcal { R } ^ { m \times n } , \boldsymbol { b } \in \mathcal { R } ^ { m }$ are a constant matrix and a vector, and $\boldsymbol { x } \in \mathcal { R } ^ { n }$ is an unknown vector. Answer the following questions.
(1) An $m \times ( n + 1 )$ matrix $\overline { \boldsymbol { A } } = ( \boldsymbol { A } \mid \boldsymbol { b } )$ is made by adding a column vector after the last column of matrix $\boldsymbol { A }$. In the case of $\boldsymbol { A } = \left( \begin{array} { c c c } 1 & 0 & - 1 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \end{array} \right)$ and $\boldsymbol { b } = \left( \begin{array} { c } 2 \\ 4 \\ 2 \end{array} \right)$, $\overline { \boldsymbol { A } } = \left( \begin{array} { c c c c } 1 & 0 & - 1 & 2 \\ 1 & 1 & 0 & 4 \\ 0 & 1 & 1 & 2 \end{array} \right)$ is obtained. Let the $i$-th column vector of the matrix $\overline { \boldsymbol { A } }$ be $\boldsymbol { a } _ { i } ( i = 1,2,3,4 )$.
(i) Find the maximum number of linearly independent vectors among $\boldsymbol { a } _ { 1 } , \boldsymbol { a } _ { 2 }$ and $\boldsymbol { a } _ { 3 }$.
(ii) Show that $a _ { 4 }$ can be represented as a linear sum of $a _ { 1 } , a _ { 2 }$ and $a _ { 3 }$, by obtaining scalars $x _ { 1 }$ and $x _ { 2 }$ that satisfy $\boldsymbol { a } _ { 4 } = x _ { 1 } \boldsymbol { a } _ { 1 } + x _ { 2 } \boldsymbol { a } _ { 2 } + \boldsymbol { a } _ { 3 }$.
(iii) Find the maximum number of linearly independent vectors among $\boldsymbol { a } _ { 1 } , \boldsymbol { a } _ { 2 } , \boldsymbol { a } _ { 3 }$ and $a _ { 4 }$.
(2) Show that the solution of the simultaneous linear equation exists when $\operatorname { rank } \overline { \boldsymbol { A } } = \operatorname { rank } \boldsymbol { A }$, for arbitrary $m , n , \boldsymbol { A }$ and $\boldsymbol { b }$.
(3) There is no solution when $\operatorname { rank } \overline { \boldsymbol { A } } > \operatorname { rank } \boldsymbol { A }$. When $m > n , \operatorname { rank } \boldsymbol { A } = n$ and $\operatorname { rank } \overline { \boldsymbol { A } } > \operatorname { rank } \boldsymbol { A }$, obtain $\boldsymbol { x }$ that minimizes the squared norm of the difference between the left hand side and the right hand side of the simultaneous linear equation, namely $\| \boldsymbol { b } - \boldsymbol { A } \boldsymbol { x } \| ^ { 2 }$.
(4) When $m < n$ and $\operatorname { rank } \boldsymbol { A } = m$, there exist multiple solutions for the simultaneous linear equation for arbitrary $\boldsymbol { b }$. Obtain $\boldsymbol { x }$ that minimizes $\| \boldsymbol { x } \| ^ { 2 }$ among them, by adopting the method of Lagrange multipliers and using the simultaneous linear equation as the constraint condition.
(5) Show that there exists a unique $\boldsymbol { P } \in \mathcal { R } ^ { n \times m }$ that satisfies the following four equations for arbitrary $m , n$ and $\boldsymbol { A }$.
$$\begin{array} { r } \boldsymbol { A P A } = \boldsymbol { A } \\ \boldsymbol { P A P } = \boldsymbol { P } \\ ( \boldsymbol { A P } ) ^ { T } = \boldsymbol { A P } \\ ( \boldsymbol { P A } ) ^ { T } = \boldsymbol { P A } \end{array}$$
(6) Show that both $\boldsymbol { x }$ obtained in (3) and $\boldsymbol { x }$ obtained in (4) are represented in the form of $x = P b$.

Answer the following questions concerning the matrix $\boldsymbol{A}$ given by
$$A = \left( \begin{array}{ccc} 0 & 3 & 0 \\ -3 & 0 & 4 \\ 0 & -4 & 0 \end{array} \right)$$
In the following, $\boldsymbol{I}$ is the $3 \times 3$ identity matrix, $\boldsymbol{O}$ is the $3 \times 3$ zero matrix, $n$ is an integer greater than or equal to 0 and $t$ is a real number.

1. Obtain all eigenvalues of the matrix $\boldsymbol{A}$.
2. Find coefficients $a, b$ and $c$ of the following equation satisfied by $\boldsymbol{A}$: $$\boldsymbol{A}^3 + a\boldsymbol{A}^2 + b\boldsymbol{A} + c\boldsymbol{I} = \boldsymbol{O}$$
3. Obtain $\boldsymbol{A}^{2n+1}$.
4. Since Equation (2) is satisfied, the following equation holds: $$\exp(t\boldsymbol{A}) = p\boldsymbol{A}^2 + q\boldsymbol{A} + r\boldsymbol{I}$$ Express coefficients $p, q$ and $r$ in terms of $t$ without using the imaginary unit.

Real symmetric matrices $\boldsymbol{A}$ and $\boldsymbol{B}$ are defined as follows:
$$\begin{aligned} & \boldsymbol{A} = \left( \begin{array}{ccc} 7 & -2 & 1 \\ -2 & 10 & -2 \\ 1 & -2 & 7 \end{array} \right), \\ & \boldsymbol{B} = \left( \begin{array}{ccc} 5 & -1 & -1 \\ -1 & 5 & -1 \\ -1 & -1 & 5 \end{array} \right). \end{aligned}$$
1. Obtain $\boldsymbol{AB}$.
Matrices $\boldsymbol{A}$ and $\boldsymbol{B}$ defined as Equations (1) and (2) satisfy $\boldsymbol{AB} = \boldsymbol{BA}$.
2. In general, two real symmetric matrices that are commutative for multiplication are simultaneously diagonalizable. Prove this for the case where all the eigenvalues are mutually different.
3. Suppose a three-dimensional real vector $\boldsymbol{v}$ whose norm is 1 is an eigenvector of $\boldsymbol{A}$ in Equation (1) corresponding to an eigenvalue $a$ as well as an eigenvector of $\boldsymbol{B}$ in Equation (2) corresponding to an eigenvalue $b$. That is, $\boldsymbol{Av} = a\boldsymbol{v}$, $\boldsymbol{Bv} = b\boldsymbol{v}$, and $\|\boldsymbol{v}\| = 1$. Obtain all the sets of $(\boldsymbol{v}, a, b)$.

Answer the following questions concerning the curved surface given by Equation (3) in the Cartesian coordinate system $xyz$. Note that $\boldsymbol{m}^{\mathrm{T}}$ indicates transpose of $\boldsymbol{m}$.
$$f(x, y, z) = 2\left(x^{2} + y^{2} + z^{2}\right) + 4yz + \frac{z - y}{\sqrt{2}} = 0 \tag{3}$$
1. When the function $f(x, y, z)$ is expressed in the following form, derive the real symmetric matrix $\boldsymbol{A}$ of order 3 and the vector $\boldsymbol{b} = \left(\begin{array}{l} b_{1} \\ b_{2} \\ b_{3} \end{array}\right)$:
$$f(x, y, z) = \left(\begin{array}{lll} x & y & z \end{array}\right) \boldsymbol{A} \left(\begin{array}{l} x \\ y \\ z \end{array}\right) + 2\boldsymbol{b}^{\mathrm{T}} \left(\begin{array}{l} x \\ y \\ z \end{array}\right)$$
2. Suppose that the matrix $\boldsymbol{A}$ derived in Question II.1 is diagonalized as $\boldsymbol{A} = \boldsymbol{P}^{\mathrm{T}}\boldsymbol{D}\boldsymbol{P}$ using an orthogonal matrix $\boldsymbol{P}$ of order 3 and a diagonal matrix $\boldsymbol{D}$, which is given by Equation (5):
$$\boldsymbol{D} = \left(\begin{array}{ccc} d_{1} & 0 & 0 \\ 0 & d_{2} & 0 \\ 0 & 0 & d_{3} \end{array}\right) \tag{5}$$
Obtain a set of $\boldsymbol{P}$ and $\boldsymbol{D}$ satisfying $d_{1} \geq d_{2} \geq d_{3}$.
3. Express the function $f$ using $X$, $Y$, and $Z$, obtained by applying the coordinate transformation defined by $\left(\begin{array}{l} X \\ Y \\ Z \end{array}\right) = \boldsymbol{P} \left(\begin{array}{l} x \\ y \\ z \end{array}\right)$, using $\boldsymbol{P}$ derived in Question II.2.
4. Consider a region surrounded by the curved surface given by Equation (3) and a plane defined by $y - z - \sqrt{2} = 0$. Obtain the volume of this region.

$$\left|\begin{array}{rrr} 2 & -3 & 2 \\ 1 & 2 & 0 \\ 2 & 3 & 0 \end{array}\right|$$
What is the value of this determinant?
A) $-1$
B) $-2$
C) $-3$
D) $-4$
E) $-6$

$$\begin{array}{r} 2x + 2y - z = 1 \\ x + y + z = 2 \\ y - z = 1 \end{array}$$
In the solution of the system of equations above, what is $x$?
A) $-3$
B) $-2$
C) $-1$
D) $0$
E) $3$