For two matrices $A = \left( \begin{array} { l l } 2 & 0 \\ 1 & 0 \end{array} \right) , B = \left( \begin{array} { r r } a & 0 \\ 2 & - 3 \end{array} \right)$, when the sum of all components of matrix $A + B$ is 6, what is the value of $a$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

For two matrices $A = \left( \begin{array} { l l } 2 & 0 \\ 1 & 1 \end{array} \right) , B = \left( \begin{array} { l l } 1 & 1 \\ 0 & 1 \end{array} \right)$, what is the sum of all components of matrix $A + 2B$? [2 points]

A graph and the matrix representing the connection relationships between each vertex of the graph are as follows. What is the value of $a + b + c + d + e$?
A B C D E
$$\left( \begin{array} { l l l l l } 0 & 1 & 1 & 0 & a \\ 1 & 0 & 1 & b & 1 \\ 1 & 1 & c & 1 & 0 \\ 0 & d & 1 & 0 & 1 \\ e & 1 & 0 & 1 & 0 \end{array} \right)$$
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

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

For the system of linear equations in $x , y$:
$$\left( \begin{array} { l l } 5 & a \\ a & 3 \end{array} \right) \binom { x } { y } = \binom { x + 5 y } { 6 x + y }$$
Find the sum of all real values of $a$ such that the system has a solution other than $x = 0 , y = 0$. [4 points]

For two matrices $A = \left( \begin{array} { l l } 1 & 1 \\ 0 & 2 \end{array} \right) , B = \left( \begin{array} { l l } 1 & 1 \\ 3 & 0 \end{array} \right)$, what is the sum of all components of the matrix $A + B$? [2 points]
(1) 5
(2) 6
(3) 7
(4) 8
(5) 9

For two matrices $A = \left( \begin{array} { l l } 1 & 1 \\ 0 & 2 \end{array} \right) , B = \left( \begin{array} { l l } 1 & 1 \\ 3 & 0 \end{array} \right)$, what is the sum of all components of the matrix $A + B$? [2 points]
(1) 5
(2) 6
(3) 7
(4) 8
(5) 9

In the following graph, how many zeros are there among the components of the matrix representing the connection relationships between vertices? [3 points]
(1) 9
(2) 11
(3) 13
(4) 15
(5) 17

Two square matrices $A , B$ satisfy $$A ^ { 2 } - A B = 3 E , \quad A ^ { 2 } B - B ^ { 2 } A = A + B$$ From the statements below, select all correct ones. (Here, $E$ is the identity matrix.) [4 points]
Statements ᄀ. The inverse matrix of $A$ exists. ㄴ. $A B = B A$ ㄷ. $( A + 2 B ) ^ { 2 } = 24 E$
(1) ᄀ
(2) ㄷ
(3) ᄀ, ㄴ
(4) ㄴ, ㄷ
(5) ᄀ, ㄴ, ㄷ

Two $2 \times 2$ square matrices $A$ and $B$ satisfy $$A ^ { 2 } - A B = 3 E , \quad A ^ { 2 } B - B ^ { 2 } A = A + B$$ Among the statements in the given options, which are correct? (Here, $E$ is the identity matrix.) [4 points]
ᄀ. The inverse matrix of $A$ exists. ㄴ. $A B = B A$ ㄷ. $( A + 2 B ) ^ { 2 } = 24 E$
(1) ᄀ
(2) ᄃ
(3) ᄀ, ᄂ
(4) ᄂ, ᄃ
(5) ᄀ, ᄂ, ᄃ

For two matrices $A = \left( \begin{array} { l l } 2 & 1 \\ 5 & 0 \end{array} \right) , B = \left( \begin{array} { l l } 1 & 0 \\ 1 & 1 \end{array} \right)$, what is the sum of all components of matrix $A - B$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

For two matrices $A = \left( \begin{array} { l l } a & 3 \\ 0 & 1 \end{array} \right) , B = \left( \begin{array} { r r } 4 & 1 \\ - 1 & 0 \end{array} \right)$, when the sum of all entries of matrix $A + B$ is 9, what is the value of $a$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

In the following graph, how many 1's are there among the components of the matrix representing the connection relationships between vertices? [3 points]
(1) 10
(2) 14
(3) 18
(4) 22
(5) 26

Two square matrices $A$ and $B$ satisfy $$A + B = ( B A ) ^ { 2 } , \quad A B A = B + E$$ Among the following statements, which are correct? (Here, $E$ is the identity matrix.) [4 points]
Statements ㄱ. $A = B ^ { 2 }$ ㄴ. $B ^ { - 1 } = A ^ { 2 } + E$ ㄷ. $A ^ { 5 } - B ^ { 5 } = A + B$
(1) ㄱ
(2) ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

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

21. Elective 4-2: Matrices and Transformations
This problem mainly tests basic knowledge of matrices and inverse matrices, tests computational ability, and tests transformation and conversion ideas. Full marks: 7 points.
Solution: (1) Since $|A| = 2 \times 1 - (-1) \times 4 = 2$,
we have $A^{-1} = \begin{pmatrix} \frac{3}{2} & -\frac{1}{2} \\ -2 & 1 \end{pmatrix}$.
(2) From $AC = B$ we get $(A^{-1}A)C = A^{-1}B$,
thus $C = A^{-1}B = \begin{pmatrix} \frac{3}{2} & -\frac{1}{2} \\ -2 & 1 \end{pmatrix} \begin{pmatrix} 1 & 1 \\ 0 & -1 \end{pmatrix} = \begin{pmatrix} \frac{3}{2} & 2 \\ -2 & -3 \end{pmatrix}$.
Elective 4-4: Coordinate Systems and Parametric Equations
This problem mainly tests basic knowledge of conversion between polar and rectangular coordinates and parametric equations of circles, tests computational ability, and tests transformation and conversion ideas. Full marks: 7 points.
Solution: (1) Eliminating the parameter $t$, we obtain the ordinary equation of the circle: $(x-1)^2 + (y+2)^2 = 9$.
From $\sqrt{2}r\sin\left(q - \frac{p}{4}\right) = m$, we get $r\sin q - r\cos q - m = 0$,
so the rectangular coordinate equation of line $l$ is $x - y - m = 0$.
(2) According to the problem, the distance from center $C$ to line $l$ equals 2, i.e.,
$$\frac{|1 - (-2) - m|}{\sqrt{2}} = 2,$$
solving we get $m = -3 \pm 2\sqrt{2}$.
Elective 4-5: Inequalities
This problem mainly tests basic knowledge of absolute value inequalities and Cauchy inequality, tests reasoning and proof ability, and tests transformation and conversion ideas. Full marks: 7 points.
Solution: (1) Since $f(x) = |x+a| + |x+b| + c \geq |(

Given the determinant $\left| \begin{array} { l l l } 1 & a & c \\ 2 & d & b \\ 3 & 0 & 0 \end{array} \right| = 6$, find the determinant $\left| \begin{array} { l l } a & c \\ d & b \end{array} \right| =$ $\_\_\_\_$

We consider the matrix with real coefficients $C \in \mathscr { M } _ { 7 } ( \mathbb { R } )$
$$C = \left( \begin{array} { l l l l l l l } 0 & 0 & \mathbf { 1 } & \mathbf { 1 } & 0 & 0 & 0 \\ 0 & \mathbf { 1 } & 0 & 0 & \mathbf { 1 } & 0 & 0 \\ \mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\ \mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\ \mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \mathbf { 1 } & 0 & 0 & \mathbf { 1 } & 0 & 0 \\ 0 & 0 & \mathbf { 1 } & \mathbf { 1 } & 0 & 0 & 0 \end{array} \right)$$
We denote $(e_1, e_2, e_3, e_4, e_5, e_6, e_7)$ the canonical basis of $\mathbb{R}^7$, and $c$ the endomorphism of $\mathbb{R}^7$ whose matrix in the canonical basis is $C$. We denote $f_1, f_2, f_3, f_4, f_5, f_6, f_7$ the column vectors of the matrix $C$.
Determine a basis of the kernel and a basis of the image of $c$, as well as the rank of $c$.

We consider the matrix with real coefficients $C \in \mathscr { M } _ { 7 } ( \mathbb { R } )$
$$C = \left( \begin{array} { l l l l l l l } 0 & 0 & \mathbf { 1 } & \mathbf { 1 } & 0 & 0 & 0 \\ 0 & \mathbf { 1 } & 0 & 0 & \mathbf { 1 } & 0 & 0 \\ \mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\ \mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\ \mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \mathbf { 1 } & 0 & 0 & \mathbf { 1 } & 0 & 0 \\ 0 & 0 & \mathbf { 1 } & \mathbf { 1 } & 0 & 0 & 0 \end{array} \right)$$
We denote $c$ the endomorphism of $\mathbb{R}^7$ whose matrix in the canonical basis is $C$, and $f_1, f_2, f_3, f_4, f_5, f_6, f_7$ the column vectors of the matrix $C$. We denote $F$ the vector subspace of $\mathbb{R}^7$ spanned by the first three column vectors $f_1, f_2$ and $f_3$ of $C$.
I.B.1) Show that $F$ is stable under $c$. I.B.2) Show that $(f_1, f_2, f_3)$ is a basis of $F$, and calculate the matrix $\Phi$ in this basis of the endomorphism $\varphi$ of $F$ induced by $c$.

We consider the matrix with real coefficients $C \in \mathscr { M } _ { 7 } ( \mathbb { R } )$
$$C = \left( \begin{array} { l l l l l l l } 0 & 0 & \mathbf { 1 } & \mathbf { 1 } & 0 & 0 & 0 \\ 0 & \mathbf { 1 } & 0 & 0 & \mathbf { 1 } & 0 & 0 \\ \mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\ \mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\ \mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \mathbf { 1 } & 0 & 0 & \mathbf { 1 } & 0 & 0 \\ 0 & 0 & \mathbf { 1 } & \mathbf { 1 } & 0 & 0 & 0 \end{array} \right)$$
and $c$ the endomorphism of $\mathbb{R}^7$ whose matrix in the canonical basis is $C$.
Notation: if $f$ is a function of class $C^1$ from an open set $\mathscr{U}$ of $\mathbb{R}^d$ ($d \geqslant 1$) to $\mathbb{R}$, we denote, for every integer $i$ such that $1 \leqslant i \leqslant d$, $\partial_i f$ the partial derivative of $f$ with respect to its $i$-th variable.
In this section, we propose to study functions $f$ of class $C^1$ from $\mathbb{R}^7$ to $\mathbb{R}$ that satisfy the condition $f \circ c = f$, that is, such that $$f\left(x_3 + x_4, x_2 + x_5, x_1, x_1, x_1, x_2 + x_5, x_3 + x_4\right) = f\left(x_1, x_2, x_3, x_4, x_5, x_6, x_7\right)$$ for all $(x_1, x_2, x_3, x_4, x_5, x_6, x_7) \in \mathbb{R}^7$.
I.E.1) What structure does the set $\mathscr{S}$ of functions $f$ of class $C^1$ from $\mathbb{R}^7$ to $\mathbb{R}$ such that $f \circ c = f$ possess? I.E.2) Show that such a function satisfies $f \circ c^n = f$ for every integer $n \geqslant 1$. I.E.3) Let $f \in \mathscr{S}$. Calculate the Jacobian matrix of $f \circ c$ at $X = (x_1, x_2, x_3, x_4, x_5, x_6, x_7)$. Deduce a system of equations relating the partial derivatives $\partial_1 f(X), \ldots, \partial_7 f(X)$ of $f$ at a point $X$ of $\mathbb{R}^7$. I.E.4) For $f \in \mathscr{S}$, calculate the Jacobian matrix of $f \circ c^2$ at $X = (x_1, x_2, x_3, x_4, x_5, x_6, x_7)$. Complete the system of equations relating the partial derivatives $\partial_1 f(X), \ldots, \partial_7 f(X)$ of $f$ at a point $X$ of $\mathbb{R}^7$ obtained in the previous question. I.E.5) Application: without further calculation, determine the linear forms $f$ on $\mathbb{R}^7$ that belong to $\mathscr{S}$.

Let $e = (e_1, \ldots, e_n)$ be a basis of $E$. We denote by $e^* = (e_1^*, \ldots, e_n^*)$ the dual basis of $e$.
a) Show that the matrix of $h$ in the bases $e$ and $e^*$ is: $$\operatorname{mat}(h, e, e^*) = \left(\varphi(e_i, e_j)\right)_{\substack{1 \leq i \leq n \\ 1 \leq j \leq n}}$$ This latter matrix will also be called the matrix of $\varphi$ in the basis $e$ and denoted $\operatorname{mat}(\varphi, e)$.
b) Let $(x,y) \in E^2$. We denote by $X$ and $Y$ the column matrices whose coefficients are the components of $x$ and $y$ in the basis $e$. Show that $\varphi(x,y) = {}^t X \Omega Y$ where $\Omega = \operatorname{mat}(\varphi, e)$ and where ${}^t X$ denotes the row matrix obtained by transposing $X$.

For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
Let $e = (e_1, \ldots, e_n)$ be a basis of $E$. We say that $e$ is $q$-orthogonal if and only if, for all $(i,j) \in \{1,\ldots,n\}^2$ with $i \neq j$, $\varphi(e_i, e_j) = 0$.
Suppose that $e$ is simultaneously $q$-orthogonal and $q'$-orthogonal. Show that, for all $i \in \{1,\ldots,n\}$, $e_i$ is an eigenvector of $h^{-1} \circ h'$.

Show that $\mathcal { M } _ { 0 } ( n , \mathbb { K } )$ is a $\mathbb { K }$-vector space; specify its dimension.

Justify that, for every pair $(A , B)$ of elements of $\mathcal { M } ( n , \mathbb { K } )$, the matrix $[ A , B ]$ belongs to $\mathcal { M } _ { 0 } ( n , \mathbb { K } )$.