For two matrices $A = \left( \begin{array} { l l } 2 & 1 \\ 1 & 1 \end{array} \right) , B = \left( \begin{array} { r r } - 1 & - 2 \\ 1 & 0 \end{array} \right)$, what is the sum of all components of the matrix $( A + B ) A$? [2 points]
(1) 9
(2) 10
(3) 11
(4) 12
(5) 13

For two matrices $A = \left( \begin{array} { l l } 2 & 1 \\ 1 & 1 \end{array} \right) , B = \left( \begin{array} { r r } - 1 & - 2 \\ 1 & 0 \end{array} \right)$, what is the sum of all entries of the matrix $( A + B ) A$? [2 points]
(1) 9
(2) 10
(3) 11
(4) 12
(5) 13

A $2 \times 2$ square matrix $A$ has the sum of all components equal to 0 and satisfies
$$A ^ { 2 } + A ^ { 3 } = - 3 A - 3 E$$
Find the sum of all components of the matrix $A ^ { 4 } + A ^ { 5 }$. (Here, $E$ is the identity matrix.) [4 points]

For two matrices $A = \left( \begin{array} { l l } 3 & 0 \\ 0 & 3 \end{array} \right) , B = \left( \begin{array} { r r } - 1 & 1 \\ 1 & 1 \end{array} \right)$, what is the sum of all components of the matrix $A B + 2 B$? [2 points]
(1) 10
(2) 8
(3) 6
(4) 4
(5) 2

For two matrices $A = \left( \begin{array} { l l } 3 & 0 \\ 0 & 3 \end{array} \right) , B = \left( \begin{array} { r r } - 1 & 1 \\ 1 & 1 \end{array} \right)$, what is the sum of all components of the matrix $A B + 2 B$? [2 points]
(1) 10
(2) 8
(3) 6
(4) 4
(5) 2

The system of linear equations in $x$ and $y$ $$\left( \begin{array} { c c } 5 - \log _ { 2 } a & 2 \\ 3 & \log _ { 2 } a \end{array} \right) \binom { x } { y } = \binom { 0 } { 0 }$$ has a solution other than $x = 0 , y = 0$. What is the sum of all values of $a$? [3 points]
(1) 8
(2) 10
(3) 12
(4) 16
(5) 20

For a $2 \times 2$ square matrix $A$ and matrix $B = \left( \begin{array} { l l } 1 & 0 \\ 1 & 1 \end{array} \right)$ such that $( B A ) ^ { 2 } = \left( \begin{array} { l l } 1 & 1 \\ 1 & 2 \end{array} \right)$, what is the matrix $( A B ) ^ { 2 }$? [4 points]
(1) $\left( \begin{array} { l l } 1 & 1 \\ 1 & 2 \end{array} \right)$
(2) $\left( \begin{array} { l l } 2 & 1 \\ 1 & 2 \end{array} \right)$
(3) $\left( \begin{array} { l l } 2 & 1 \\ 1 & 1 \end{array} \right)$
(4) $\left( \begin{array} { l l } 1 & 2 \\ 2 & 1 \end{array} \right)$
(5) $\left( \begin{array} { l l } 1 & 1 \\ 2 & 1 \end{array} \right)$

For a $2 \times 2$ matrix $A$ and matrix $B = \left( \begin{array} { l l } 1 & 0 \\ 1 & 1 \end{array} \right)$ satisfying $( B A ) ^ { 2 } = \left( \begin{array} { l l } 1 & 1 \\ 1 & 2 \end{array} \right)$, what is the matrix $( A B ) ^ { 2 }$? [4 points]
(1) $\left( \begin{array} { l l } 1 & 1 \\ 1 & 2 \end{array} \right)$
(2) $\left( \begin{array} { l l } 2 & 1 \\ 1 & 2 \end{array} \right)$
(3) $\left( \begin{array} { l l } 2 & 1 \\ 1 & 1 \end{array} \right)$
(4) $\left( \begin{array} { l l } 1 & 2 \\ 2 & 1 \end{array} \right)$
(5) $\left( \begin{array} { l l } 1 & 1 \\ 2 & 1 \end{array} \right)$

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

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

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

A certain company converts the raw scores of applicants' reasoning ability test and spatial perception ability test for use. When the raw score of the reasoning ability test is $x$ and the raw score of the spatial perception ability test is $y$, the two converted scores $p$ and $q$ are as follows. $$\binom { p } { q } = \left( \begin{array} { l l } 3 & 2 \\ 2 & 3 \end{array} \right) \binom { x } { y }$$ When the converted scores of applicants A, B, and C are as shown in the table, let the raw scores of applicants A, B, and C on the reasoning ability test be $a , b , c$, respectively. Which of the following correctly represents the order of magnitude of $a , b , c$? [4 points]

Converted Score / Test Taker	A	B	C
$p$	45	50	45
$q$	40	50	50

(1) $a > b > c$
(2) $a > c > b$
(3) $b > a > c$
(4) $b > c > a$
(5) $c > b > a$

For a $2 \times 2$ square matrix $A$, the $(i,j)$ component $a_{ij}$ is $$a_{ij} = i - j \quad (i = 1,2,\ j = 1,2)$$ What is the $(2,1)$ component of the matrix $A + A^2 + A^3 + \cdots + A^{2010}$? [4 points]
(1) $-2010$
(2) $-1$
(3) $0$
(4) $1$
(5) $2010$

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

The sum of all components of the inverse matrix $A ^ { - 1 }$ of the matrix $A = \left( \begin{array} { r r } 1 & - 2 \\ 0 & 1 \end{array} \right)$ is? [2 points]
(1) 5
(2) 4
(3) 3
(4) 2
(5) 1

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

A $2 \times 2$ square matrix $A$ satisfies the following conditions. (where $E$ is the identity matrix and $O$ is the zero matrix)
(A) $A ^ { 2 } + 2 A - E = O$
(B) $A \binom { 1 } { - 1 } = \binom { 3 } { 4 }$ Find the value of $x + y$ for real numbers $x , y$ satisfying $( A + 2 E ) \binom { x } { y } = \binom { 3 } { - 3 }$. [4 points]

For two matrices $A = \left( \begin{array} { l l } 0 & 0 \\ 1 & 1 \end{array} \right) , B = \left( \begin{array} { l l } 1 & 0 \\ 1 & 1 \end{array} \right)$, what is the sum of all entries of the matrix $2A + B$? [2 points]
(1) 10
(2) 9
(3) 8
(4) 7
(5) 6

For two matrices $A = \left( \begin{array} { l l } 0 & 0 \\ 1 & 1 \end{array} \right) , B = \left( \begin{array} { l l } 1 & 0 \\ 1 & 1 \end{array} \right)$, what is the sum of all components of the matrix $2 A + B$? [2 points]
(1) 10
(2) 9
(3) 8
(4) 7
(5) 6

What is the sum of all entries of the matrix representing the connection relationships between vertices of the following graph? [3 points]
(1) 6
(2) 8
(3) 10
(4) 12
(5) 14

For the system of equations in $x, y$: $$\left( \begin{array} { c c } a + 1 & a \\ 1 & 1 \end{array} \right) \binom{x}{y} = \binom{-4}{1}$$ the solution satisfies the equation $x + 2y - 4a = 0$. What is the value of the constant $a$?
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

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

Two $2 \times 2$ square matrices $A , B$ satisfy
$$2 A ^ { 2 } + A B = E , \quad A B + B A = 2 A + E$$
Which of the following statements are correct? Choose all that apply from . (Given that $E$ is the identity matrix.) [4 points]

ㄱ. $A ^ { - 1 } = 2 A + B$ ㄴ. $B = 2 A + 2 E$ ㄷ. $( B - E ) ^ { 2 } = O$ (where $O$ is the zero matrix.)
(1) ㄴ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄱ, ㄷ
(5) ㄱ, ㄴ, ㄷ