Show that, for all natural integers $p$ and $q$ greater than or equal to 2, for any matrix $M = (M(i,j))_{1 \leqslant i,j \leqslant q} \in \mathcal{M}_q(\mathbb{R})$ and for all $(i,j) \in \llbracket 1,q \rrbracket^2$, the coefficient with index $(i,j)$ of the matrix $M^p$ is $$\sum_{(k_2,\ldots,k_p) \in \llbracket 1,q \rrbracket^{p-1}} M(i,k_2)\left(\prod_{r=2}^{p-1} M(k_r, k_{r+1})\right) M(k_p, j),$$ the product being equal to 1 in the case where $p = 2$.

Let $A$ and $B$ be two $3 \times 3$ matrices such that $( A + B ) ^ { 2 } = A ^ { 2 } + B ^ { 2 }$. Which of the following must be true?
(A) $A$ and $B$ are zero matrices.
(B) $A B$ is the zero matrix.
(C) $( A - B ) ^ { 2 } = A ^ { 2 } - B ^ { 2 }$
(D) $( A - B ) ^ { 2 } = A ^ { 2 } + B ^ { 2 }$

Let $P = \left[ \begin{array} { c c c } 1 & 0 & 0 \\ 4 & 1 & 0 \\ 16 & 4 & 1 \end{array} \right]$ and $I$ be the identity matrix of order 3. If $Q = \left[ q _ { i j } \right]$ is a matrix such that $P ^ { 50 } - Q = I$, then $\frac { q _ { 31 } + q _ { 32 } } { q _ { 21 } }$ equals
(A) 52
(B) 103
(C) 201
(D) 205

Let $M$ be a $3 \times 3$ invertible matrix with real entries and let $I$ denote the $3 \times 3$ identity matrix. If $M ^ { - 1 } = \operatorname { adj } ( \operatorname { adj } M )$, then which of the following statements is/are ALWAYS TRUE?
(A) $M = I$
(B) $\operatorname { det } M = 1$
(C) $M ^ { 2 } = I$
(D) $( \operatorname { adj } M ) ^ { 2 } = I$

Consider the matrix
$$P = \left( \begin{array} { l l l } 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{array} \right)$$
Let the transpose of a matrix $X$ be denoted by $X ^ { T }$. Then the number of $3 \times 3$ invertible matrices $Q$ with integer entries, such that
$$Q ^ { - 1 } = Q ^ { T } \text { and } P Q = Q P$$
is

(A)

32

(B)

8

(C)

16

(D)

24

If $B$ is a $3 \times 3$ matrix such that $B ^ { 2 } = 0$, then det. $\left[ ( I + B ) ^ { 50 } - 50 B \right]$ is equal to:
(1) 1
(2) 2
(3) 3
(4) 50

If $A = \begin{pmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b \end{pmatrix}$ is a matrix satisfying the equation $AA^T = 9I$, where $I$ is $3 \times 3$ identity matrix, then the ordered pair $(a, b)$ is equal to:
(1) $(2, -1)$
(2) $(-2, 1)$
(3) $(2, 1)$
(4) $(-2, -1)$

$A = \left[ \begin{array} { c c c } 1 & 2 & 2 \\ 2 & 1 & - 2 \\ a & 2 & b \end{array} \right]$ is a matrix satisfying the equation $A A ^ { T } = 9 I$, where $I$ is $3 \times 3$ identity matrix, then the ordered pair $( a , b )$ is equal to
(1) $( - 2 , - 1 )$
(2) $( 2 , - 1 )$
(3) $( - 2,1 )$
(4) $( 2,1 )$

Let $A = \begin{pmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{pmatrix}$, $\alpha \in R$ such that $A^{32} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$. Then, a value of $\alpha$ is:
(1) 0
(2) $\frac{\pi}{16}$
(3) $\frac{\pi}{64}$
(4) $\frac{\pi}{32}$

Let $a , b , c \in R$ be all non-zero and satisfies $a ^ { 3 } + b ^ { 3 } + c ^ { 3 } = 2$. If the matrix $A = \left[ \begin{array} { c c c } a & b & c \\ b & c & a \\ c & a & b \end{array} \right]$ satisfies $A ^ { T } A = I$, then a value of $a b c$ can be
(1) $- \frac { 1 } { 3 }$
(2) $\frac { 1 } { 3 }$
(3) 3
(4) $\frac { 2 } { 3 }$

Let $A = \left[ \begin{array} { l l l } x & y & z \\ y & z & x \\ z & x & y \end{array} \right]$, where $x , y$ and $z$ are real numbers such that $x + y + z > 0$ and $x y z = 2$. If $A ^ { 2 } = I _ { 3 }$, then the value of $x ^ { 3 } + y ^ { 3 } + z ^ { 3 }$ is

Let $P = \left[ \begin{array} { c c c } - 30 & 20 & 56 \\ 90 & 140 & 112 \\ 120 & 60 & 14 \end{array} \right]$ and $A = \left[ \begin{array} { c c c } 2 & 7 & \omega ^ { 2 } \\ - 1 & - \omega & 1 \\ 0 & - \omega & - \omega + 1 \end{array} \right]$ where $\omega = \frac { - 1 + i \sqrt { 3 } } { 2 }$, and $I _ { 3 }$ be the identity matrix of order 3 . If the determinant of the matrix $\left( P ^ { - 1 } A P - I _ { 3 } \right) ^ { 2 }$ is $\alpha \omega ^ { 2 }$, then the value of $\alpha$ is equal to $\_\_\_\_$.

Let $\alpha$ and $\beta$ be real numbers. Consider a $3 \times 3$ matrix $A$ such that $A ^ { 2 } = 3 A + \alpha I$. If $A ^ { 4 } = 21 A + \beta I$, then
(1) $\alpha = 1$
(2) $\alpha = 4$
(3) $\beta = 8$
(4) $\beta = - 8$

Let $R = \begin{pmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{pmatrix}$ 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 entries of $M$. Then, among the statements: I. Trace$(R) = 0$ II. If Trace$(\operatorname{adj}(\operatorname{adj}(R))) = 0$, then $R$ has exactly one non-zero entry.
(1) Both (I) and (II) are true
(2) Only (II) is true
(3) Neither (I) nor (II) is true
(4) Only (I) is true

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) $- 12$

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.