A student recorded the bimonthly grades of some of his subjects in a table. He observed that the numerical entries in the table formed a $4 \times 4$ matrix, and that he could calculate the annual averages of these subjects using matrix multiplication. All tests had the same weight, and the table he obtained is shown below.

	$1^{st}$ bimonth	$2^{nd}$ bimonth	$3^{rd}$ bimonth	$4^{th}$ bimonth
Mathematics	5.9	6.2	4.5	5.5
Portuguese	6.6	7.1	6.5	8.4
Geography	8.6	6.8	7.8	9.0
History	6.2	5.6	5.9	7.7

To obtain these averages, he multiplied the matrix obtained from the table by
(A) $\left[\frac{1}{2}\quad\frac{1}{2}\quad\frac{1}{2}\quad\frac{1}{2}\right]$

For square matrices $A$ and $B$ of order 2, select all statements that are always true from . (Here, $E$ is the identity matrix and $O$ is the zero matrix.) [3 points]
ㄱ. $( A + B ) ^ { 2 } = A ^ { 2 } + 2 A B + B ^ { 2 }$ ㄴ. If $A ^ { 2 } + A - 2 E = O$, then $A$ has an inverse matrix. ㄷ. If $A \neq O$ and $A ^ { 2 } = A$, then $A = E$.
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄱ, ㄴ
(5) ㄴ, ㄷ

The following table shows the manufacturing cost per unit, selling price, and sales volume for two products A and B produced by a company last year.

Category	Product A	Product B
Manufacturing Cost	$a _ { 11 }$	$a _ { 12 }$
Selling Price	$a _ { 21 }$	$a _ { 22 }$

Sales Volume	First Half	Second Half
A	$b _ { 11 }$	$b _ { 12 }$
B	$b _ { 21 }$	$b _ { 22 }$

Represent the above tables as matrices $A = \left( \begin{array} { l l } a _ { 11 } & a _ { 12 } \\ a _ { 21 } & a _ { 22 } \end{array} \right)$ and $B = \left( \begin{array} { l l } b _ { 11 } & b _ { 12 } \\ b _ { 21 } & b _ { 22 } \end{array} \right)$ respectively, and let the product of these two matrices be $A B = \left( \begin{array} { l l } a & b \\ c & d \end{array} \right)$. When the profit per unit is defined as the selling price minus the manufacturing cost, select all correct statements from . [3 points]
ㄱ. $a + b$ is the total manufacturing cost of products sold in the first half of last year. ㄴ. $c + d$ is the total selling amount of products sold throughout last year. ㄷ. $d - b$ is the total profit from products sold in the second half of last year.
(1) ㄱ
(2) ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

For the quadratic equation $x ^ { 2 } - 4 x - 1 = 0$ with roots $\alpha$ and $\beta$, find the sum of all components of the product of two matrices $\left( \begin{array} { l l } \alpha & \beta \\ 0 & \alpha \end{array} \right) \left( \begin{array} { l l } \beta & \alpha \\ 0 & \beta \end{array} \right)$.

For two matrices $A = \left( \begin{array} { l l } 1 & 1 \\ 1 & 0 \end{array} \right) , B = \left( \begin{array} { l l } 1 & 2 \\ 3 & 4 \end{array} \right)$, what is the matrix $X$ that satisfies $2 A + X = A B$? [2 points]
(1) $\left( \begin{array} { r r } 1 & 5 \\ 3 & - 1 \end{array} \right)$
(2) $\left( \begin{array} { r r } 2 & 4 \\ - 1 & 2 \end{array} \right)$
(3) $\left( \begin{array} { l l } 2 & 5 \\ 7 & 0 \end{array} \right)$
(4) $\left( \begin{array} { l l } 2 & 7 \\ 4 & 5 \end{array} \right)$
(5) $\left( \begin{array} { l l } 4 & 6 \\ 1 & 2 \end{array} \right)$

For all non-zero $2 \times 2$ square matrices $A , B$ satisfying the following three conditions, which matrix is always equal to $B ^ { 3 } + 2 B A ^ { 3 }$? (Here, $E$ is the identity matrix.) [3 points] (가) $A B = B A$ (나) $( E - B ) ^ { 2 } = E - B$ (다) $A B = - B$
(1) $2 A$
(2) $- A$
(3) $E$
(4) $2 B$
(5) $- B$

For two matrices $A = \left( \begin{array} { l l } 1 & 1 \\ 1 & 0 \end{array} \right) , B = \left( \begin{array} { l l } 1 & 2 \\ 3 & 4 \end{array} \right)$, what is the matrix $X$ that satisfies $2 A + X = A B$? [2 points]
(1) $\left( \begin{array} { r r } 1 & 5 \\ 3 & - 1 \end{array} \right)$
(2) $\left( \begin{array} { r r } 2 & 4 \\ - 1 & 2 \end{array} \right)$
(3) $\left( \begin{array} { l l } 2 & 5 \\ 7 & 0 \end{array} \right)$
(4) $\left( \begin{array} { l l } 2 & 7 \\ 4 & 5 \end{array} \right)$
(5) $\left( \begin{array} { l l } 4 & 6 \\ 1 & 2 \end{array} \right)$

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

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

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

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 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 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 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

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

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

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

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

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]

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 $A = \left( \begin{array} { l l } 1 & 1 \\ 0 & 2 \end{array} \right) , B = \left( \begin{array} { l l } 1 & 1 \\ 3 & 0 \end{array} \right)$, 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