Exercise 4 — Candidates who have followed the specialisation course
For each of the following statements, indicate whether it is true or false, by justifying the answer.
One point is awarded for each correct answer that is properly justified. An unjustified answer is not taken into account. No answer is not penalised.
1. Statement 1: The solutions of the equation $7 x - 12 y = 5$, where $x$ and $y$ are relative integers, are the pairs $( - 1 + 12 k ; - 1 + 7 k )$ where $k$ ranges over the set of relative integers.
2. Statement 2: For all natural number $n$, the remainder of the Euclidean division of $4 + 3 \times 15 ^ { n }$ by 3 is equal to 1.
3. Statement 3: The equation $n \left( 2 n ^ { 2 } - 3 n + 5 \right) = 3$, where $n$ is a natural number, has at least one solution.
4. Let $t$ be a real number. We set $A = \left( \begin{array} { c c } t & 3 \\ 2 t & - t \end{array} \right)$.
Statement 4: There is no value of the real number $t$ for which $A ^ { 2 } = \left( \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right)$.
5. Consider the matrices $A = \left( \begin{array} { c c c } 0 & 1 & - 1 \\ - 1 & 2 & - 1 \\ 1 & - 1 & 2 \end{array} \right)$ and $I _ { 3 } = \left( \begin{array} { l l l } 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right)$.
Statement 5: For all integer $n \geqslant 2 , A ^ { n } = \left( 2 ^ { n } - 1 \right) A + \left( 2 - 2 ^ { n } \right) I _ { 3 }$.

There exists a real $3 \times 3$ orthogonal matrix with only non-zero entries.

For any $n \geq 2$ there is an $n \times n$ matrix $A$ with real entries such that $A ^ { 2 } = A$ and trace $( A ) = n + 1$.

Let $A \in M_{m \times n}(\mathbb{R})$ be of rank $m$. Choose the correct statement(s) from below:
(A) The map $\mathbb{R}^n \longrightarrow \mathbb{R}^m$ given by $v \mapsto Av$ is injective;
(B) There exist matrices $B \in M_m(\mathbb{R})$ and $C \in M_n(\mathbb{R})$ such that $BAC = \left[I_m \mid \mathbf{0}_{n-m}\right]$;
(C) There exist matrices $B \in \mathrm{GL}_m(\mathbb{R})$ and $C \in \mathrm{GL}_n(\mathbb{R})$ such that $BAC = \left[I_m \mid \mathbf{0}_{n-m}\right]$;
(D) For every $(B, C) \in M_m(\mathbb{R}) \times M_n(\mathbb{R})$ such that $BAC = \left[I_m \mid \mathbf{0}_{n-m}\right]$, $C$ is uniquely determined by $B$.

Let $A$ and $B$ be $5 \times 5$ real matrices with $A^{2} = B^{2}$. Which of the following statements is/are correct?
(A) Either $A = B$ or $A = -B$.
(B) $A$ and $B$ have the same eigen spaces.
(C) $A$ and $B$ have the same eigen values.
(D) $A^{13} B^{3} = A^{3} B^{13}$.

Let $A \in M _ { 2 } ( \mathbb { R } )$ be a nonzero matrix. Pick the correct statement(s) from below.
(A) If $A ^ { 2 } = 0$, then $\left( I _ { 2 } - A \right) ^ { 5 } = 0$.
(B) If $A ^ { 2 } = 0$, then ( $I _ { 2 } - A$ ) is invertible.
(C) If $A ^ { 3 } = 0$, then $A ^ { 2 } = 0$.
(D) If $A ^ { 2 } = A ^ { 3 } \neq 0$, then $A$ is invertible.

For $A \in M _ { 3 } ( \mathbb { C } )$, let $W _ { A } = \left\{ B \in M _ { 3 } ( \mathbb { C } ) \mid A B = B A \right\}$. Which of the following is/are true?
(A) For all diagonal $A \in M _ { 3 } ( \mathbb { C } )$, $W _ { A }$ is a linear subspace of $M _ { 3 } ( \mathbb { C } )$ with $\operatorname { dim } _ { \mathbb { C } } W _ { A } \geq 3$.
(B) For all $A \in M _ { 3 } ( \mathbb { C } ) , W _ { A }$ is a linear subspace of $M _ { 3 } ( \mathbb { C } )$ with $\operatorname { dim } _ { \mathbb { C } } W _ { A } > 3$.
(C) There exists $A \in M _ { 3 } ( \mathbb { C } )$ such that $W _ { A }$ is a linear subspace of $M _ { 3 } ( \mathbb { C } )$ with $\operatorname { dim } _ { \mathbb { C } } W _ { A } = 3$.
(D) If $A \in M _ { 3 } ( \mathbb { C } )$ is diagonalizable, then every element of $W _ { A }$ is diagonalizable.

Let $\mathcal { P } _ { n } = \{ f ( x ) \in \mathbb { R } [ x ] \mid \operatorname { deg } f ( x ) \leq n \}$, considered as an ($n + 1$)-dimensional real vector space. Let $T$ be the linear operator $f \mapsto f + \frac { \mathrm { d } f } { \mathrm {~d} x }$ on $\mathcal { P } _ { n }$. Pick the correct statement(s) from below.
(A) $T$ is invertible.
(B) $T$ is diagonalizable.
(C) $T$ is nilpotent.
(D) $( T - I ) ^ { 2 } = ( T - I )$ where $I$ is the identity map.

Let $A = \left[\begin{array}{ccc} 1 & 2 & 3 \\ 10 & 20 & 30 \\ 11 & 22 & k \end{array}\right]$ and $\mathbf{v} = \left[\begin{array}{l} x \\ y \\ z \end{array}\right]$, where $k$ is a constant and $x, y, z$ are variables.
Statements
(9) Regardless of the value of $k$, the matrix $A$ is not invertible, i.e., there is no $3 \times 3$ matrix $B$ such that $BA =$ the $3 \times 3$ identity matrix. (10) There is a unique $k$ such that determinant of $A$ is 0. (11) The set of solutions $(x, y, z)$ of the matrix equation $A\mathbf{v} = \left[\begin{array}{l} 0 \\ 0 \\ 0 \end{array}\right]$ is either a line or a plane containing the origin. (12) If the equation $A\mathbf{v} = \left[\begin{array}{c} p \\ q \\ r \end{array}\right]$ has a solution, then it must be true that $q = 10p$.

Let $p \geq 3$ be a prime number and $V$ be an $n$-dimensional vector space over $\mathbb { F } _ { p }$. Let $T : V \rightarrow V$ be a linear transformation. Select all the true statement(s) from below.
(A) $T$ has an eigenvalue in $\mathbb { F } _ { p }$.
(B) If $T ^ { p - 1 } = I$, then the minimal polynomial of $T$ has distinct roots in $\mathbb { F } _ { p }$.
(C) If $T \neq I$ and $T ^ { p - 1 } = I$, then the characteristic polynomial of $T$ has distinct roots in $\mathbb { F } _ { p }$.
(D) If $T ^ { p - 1 } = I$, then $T$ is diagonalizable over $\mathbb { F } _ { p }$.

6. Suppose that $A$ is an $n \times n$ matrix with $n \geq 6$. For an $n \times 1$ vector $b$ consider the equations
$$\begin{aligned} & A x = b , \text { for an } n \times 1 \text { vector } x \\ & A ^ { T } x = b \text { for an } n \times 1 \text { vector } x \end{aligned}$$
where $A ^ { T }$ is the transpose of the matrix $A$. Which of the following statements are correct?
(a) If equation (??) admits a solution for all $b$, then $A ^ { - 1 }$ exists.
(b) If equation (??) admits a solution for all $b$, then equation (??) also admits a solution for all $b$.
(c) If equation (??) admits a solution for some $b$, then $A ^ { - 1 }$ exists.
(d) If equation (??) admits a solution for some $b$, then equation (??) also admits a solution for that $b$.

8. Which of the following statements are valid for all $n \times n$ matrices $A , B$ :
(a) $\left( A ^ { T } A \right) ^ { T } = A A ^ { T }$.
(b) If $A , B$ are invertible, then inverse of $A B$ is $A ^ { - 1 } B ^ { - 1 }$.
(c) $( A + B ) ^ { T } = A ^ { T } + B ^ { T }$
(d) $A x = B x$ for some $n \times 1$ vector $x$ implies that $A y = B y$ for all $n \times 1$ vectors $y$.

Let $T : \mathbb { R } ^ { 3 } \longrightarrow \mathbb { R } ^ { 3 }$ be a linear transformation such that $T \neq 0$ and $T ^ { 4 } = 0$. Pick the correct statement(s) from below.
(A) $T ^ { 3 } = 0$.
(B) $\operatorname { Image } ( T ) \neq \operatorname { Image } \left( T ^ { 2 } \right)$.
(C) $\operatorname { rank } \left( T ^ { 2 } \right) \leq 1$.
(D) $\operatorname { rank } ( T ) = 2$.

4. Let $A = \left[ \begin{array} { l l } a & b \\ c & d \end{array} \right]$ be a real matrix which satisfies $A ^ { - 1 } = A$. Which of the following statements is/are always true?
(a) $a + d = 0$
(b) $a d - b c = 1$
(c) $a d - b c \neq 0$
(d) $a ^ { 2 } + 2 b c + d ^ { 2 } = 2$

7. Let $B = \left( \left( b _ { i , j } \right) \right)$ be an $n \times n$ matrix. Let $p : \{ 1,2 , \ldots , n \} \mapsto \{ 1,2 , \ldots , n \}$ be a bijection (i.e. a one-to-one correspondence) and let a matrix $A = \left( \left( a _ { i , j } \right) \right)$ be defined by
$$a _ { i , j } = b _ { p ( i ) , p ( j ) } , \quad 1 \leq i , j \leq n .$$
Which of the following statement(s) is/are true for all choices of $B$ and $P$.
(a) $A$ admits an inverse if and only if $B$ admits an inverse.
(b) For any $x , y \in \mathbb { R } ^ { n } , A x = y$ admits a solution if and only if $B x = y$ admits a solution.
(c) $A$ and $B$ have the same trace.
(d) $A$ and $B$ have the same eigenvectors.

Two $2 \times 2$ square matrices $A , B$ satisfy $A ^ { 2 } = E , B ^ { 2 } = B$. Which of the following statements in the given options are always true? (Note: $E$ is the identity matrix.) [3 points]
Given Options ㄱ. If matrix $B$ has an inverse matrix, then $B = E$. ㄴ. $( E - A ) ^ { 5 } = 2 ^ { 4 } ( E - A )$ ㄷ. $( E - A B A ) ^ { 2 } = E - A B A$
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

Two square matrices $A , B$ of order 2 satisfy $A ^ { 2 } = E , B ^ { 2 } = B$. Which of the following statements in are always correct? (Here, $E$ is the identity matrix.) [3 points]
Remarks ㄱ. If matrix $B$ has an inverse, then $B = E$. ㄴ. $( E - A ) ^ { 5 } = 2 ^ { 4 } ( E - A )$ ㄷ. $( E - A B A ) ^ { 2 } = E - A B A$
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

For two non-zero real numbers $a , b$, two square matrices $A , B$ satisfy $AB = \left( \begin{array} { l l } a & 0 \\ 0 & b \end{array} \right)$. Which of the following in are correct? [4 points]
ㄱ. If $a = b$, then the inverse matrix $A ^ { - 1 }$ of $A$ exists. ㄴ. If $a = b$, then $A B = B A$. ㄷ. If $a \neq b$ and $A = \left( \begin{array} { l l } 1 & 0 \\ 1 & 1 \end{array} \right)$, then $A B = B A$.
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

For two non-zero real numbers $a , b$, two square matrices $A , B$ satisfy $A B = \left( \begin{array} { l l } a & 0 \\ 0 & b \end{array} \right)$. Which of the following in are correct? [4 points]
ㄱ. If $a = b$, then the inverse matrix $A ^ { - 1 }$ of $A$ exists. ㄴ. If $a = b$, then $A B = B A$. ㄷ. If $a \neq b$ and $A = \left( \begin{array} { l l } 1 & 0 \\ 1 & 1 \end{array} \right)$, then $A B = B A$.
(1) ᄀ
(2) ᄃ
(3) ᄀ, ᄂ
(4) ㄴ, ㄷ
(5) ᄀ, ᄂ, ᄃ

Let the set $U$ be $$U = \left\{ \left. \left( \begin{array} { l l } a & b \\ c & d \end{array} \right) \right\rvert \, a , b , c , d \text{ are positive numbers not equal to } 1 \right\}$$ Let the subset $S$ of $U$ be $$S = \left\{ \left. \left( \begin{array} { l l } a & b \\ c & d \end{array} \right) \right\rvert \, \log _ { a } d = \log _ { b } c , \quad a \neq b , \quad b c \neq 1 \right\}$$ Which of the following are correct? Select all that apply from . [4 points]
ㄱ. If $A = \left( \begin{array} { c c } 4 & 9 \\ 3 & 2 \end{array} \right)$, then $A \in S$. ㄴ. If $A \in U$ and $A$ has an inverse matrix, then $A \in S$. ㄷ. If $A \in S$, then $A$ has an inverse matrix.
(1) ㄱ
(2) ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

Sets $S$ and $T$ with $1 \times 2$ matrices and $2 \times 1$ matrices as elements, respectively, are as follows. $$S = \{ ( a \; b ) \mid a + b \neq 0 \} , \quad T = \left\{ \left. \binom { p } { q } \right\rvert \, p q \neq 0 \right\}$$ For an element $A$ of set $S$, which of the following statements in are correct? [4 points]
ㄱ. For an element $P$ of set $T$, $PA$ does not have an inverse matrix. ㄴ. For an element $B$ of set $S$ and an element $P$ of set $T$, if $PA = PB$, then $A = B$. ㄷ. Among the elements of set $T$, there exists $P$ satisfying $PA \binom { 1 } { 1 } = \binom { 1 } { 1 }$.
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

Set $S$ has $1 \times 2$ matrices as elements and set $T$ has $2 \times 1$ matrices as elements, as follows. $$S = \{ ( a \; b ) \mid a + b \neq 0 \} , \quad T = \left\{ \left. \binom { p } { q } \right\rvert \, p q \neq 0 \right\}$$ Which of the following are correct for element $A$ of set $S$? Choose all that apply from $\langle$Remarks$\rangle$. [4 points]
$\langle$Remarks$\rangle$ ㄱ. For element $P$ of set $T$, $PA$ does not have an inverse matrix. ㄴ. For element $B$ of set $S$ and element $P$ of set $T$, if $PA = PB$ then $A = B$. ㄷ. Among the elements of set $T$, there exists $P$ satisfying $PA \binom { 1 } { 1 } = \binom { 1 } { 1 }$.
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

Two $2 \times 2$ square matrices $A , B$ satisfy
$$A ^ { 2 } + B = 3 E , \quad A ^ { 4 } + B ^ { 2 } = 7 E$$
Which of the following statements are correct? (where $E$ is the identity matrix) [4 points]
ㄱ. $A B = B A$ ㄴ. $B ^ { - 1 } = A ^ { 2 }$ ㄷ. $A ^ { 6 } + B ^ { 3 } = 18 E$
(1) ㄱ
(2) ㄴ
(3) ㄱ, ㄴ
(4) ㄱ, ㄷ
(5) ㄱ, ㄴ, ㄷ

Two square matrices $A , B$ satisfy
$$A ^ { 2 } + B = 3 E , \quad A ^ { 4 } + B ^ { 2 } = 7 E$$
Which of the following are correct? Choose all that apply from . (Here, $E$ is the identity matrix.) [4 points]
Remarks ㄱ. $A B = B A$ ㄴ. $B ^ { - 1 } = A ^ { 2 }$ ㄷ. $A ^ { 6 } + B ^ { 3 } = 18 E$
(1) ㄱ
(2) ㄴ
(3) ㄱ, ㄴ
(4) ㄱ, ㄷ
(5) ㄱ, ㄴ, ㄷ

Two square matrices $A , B$ satisfy
$$A B + A ^ { 2 } B = E , \quad ( A - E ) ^ { 2 } + B ^ { 2 } = O$$
Which of the following statements in the given options are correct? (Here, $E$ is the identity matrix and $O$ is the zero matrix.) [4 points]
Options
$\text{ᄀ}$. The inverse matrix of $B$ exists. $\text{ㄴ}$. $A B = B A$ $\text{ㄷ}$. $\left( A ^ { 3 } - A \right) ^ { 2 } + E = O$
(1) ㄴ
(2) ㄷ
(3) ᄀ, ㄴ
(4) ᄀ, ㄷ
(5) ᄀ, ㄴ, ㄷ