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cmi-entrance 2025 Q4 True/False or Multiple-Select Conceptual Reasoning View
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) $a ^ { 2 } + 2 b c + d ^ { 2 } = 2$
cmi-entrance 2025 Q7 True/False or Multiple-Select Conceptual Reasoning View
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.
cmi-entrance 2025 Q13 Determinant and Rank Computation View
13. Let $n$ be an integer, $n \geq 4$. $A$ is an $n \times n$ matrix with real entries. The matrix $B$ is obtained by the following sequence of operations on $A$. First, multiply each entry of $A$ by 2 . Then add 3 times the second column to the third column. Finally, swap the first and the fourth columns. If $\operatorname { det } ( A ) = 5$, which of the following statements are true?
(a) 10 divides $\operatorname { det } ( B )$
(b) $\operatorname { det } ( B ) = - 5$
(c) 100 divides $\operatorname { det } ( B )$
(d) $\operatorname { det } ( B ) = - 2 ^ { n } \cdot 5$
csat-suneung 2005 Q2 2 marks Linear System and Inverse Existence View
For two matrices $A = \left( \begin{array} { l l } 1 & 2 \\ 2 & 5 \end{array} \right) , B = \left( \begin{array} { l l } 2 & - 3 \\ 1 & - 2 \end{array} \right)$, what is the sum of all components of matrix $X$ that satisfies $A X = B$? [2 points]
(1) $- 2$
(2) $- 1$
(3) 0
(4) 1
(5) 2
csat-suneung 2005 Q2 2 marks Linear System and Inverse Existence View
For two matrices $A = \left( \begin{array} { l l } 1 & 2 \\ 2 & 5 \end{array} \right) , B = \left( \begin{array} { l l } 2 & - 3 \\ 1 & - 2 \end{array} \right)$, what is the sum of all components of matrix $X$ that satisfies $A X = B$? [2 points]
(1) - 2
(2) - 1
(3) 0
(4) 1
(5) 2
csat-suneung 2005 Q6 3 marks Matrix Algebra and Product Properties View
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
csat-suneung 2005 Q18 Matrix Algebra and Product Properties View
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 $\left( \begin{array} { l l } \alpha & \beta \\ 0 & \alpha \end{array} \right) \left( \begin{array} { l l } \beta & \alpha \\ 0 & \beta \end{array} \right)$.
csat-suneung 2005 Q22 4 marks Matrix Entry and Coefficient Identities View
Natural numbers are arranged at regular intervals on the sides and vertices of squares with side lengths $1, 3, 5, \cdots, 2 n - 1, \cdots$ as shown in the figure below. In each square, 1 is placed directly above the lower left vertex.
Let the $2 \times 2$ matrices with the natural numbers at the four vertices of each square as components be $A _ { 1 } , A _ { 2 } , A _ { 3 } , \cdots , A _ { n } , \cdots$ in order. For example, $A _ { 1 } = \left( \begin{array} { l l } 1 & 2 \\ 4 & 3 \end{array} \right) , A _ { 2 } = \left( \begin{array} { c c } 3 & 6 \\ 12 & 9 \end{array} \right)$. Find the sum of all components of matrix $A _ { 15 }$. [4 points]
csat-suneung 2006 Q2 2 marks Matrix Algebra and Product Properties View
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) $\left( \begin{array} { l l } 4 & 6 \\ 1 & 2 \end{array} \right)$
csat-suneung 2006 Q2 2 marks Matrix Algebra and Product Properties View
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) $\left( \begin{array} { l l } 4 & 6 \\ 1 & 2 \end{array} \right)$
csat-suneung 2006 Q6 3 marks Matrix Algebra and Product Properties View
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) $- B$
csat-suneung 2006 Q12 4 marks Circle-Line Intersection and Point Conditions View
For two points $\mathrm { A } ( 1 , \sqrt { 3 } ) , \mathrm { B } ( 1 , - \sqrt { 3 } )$ on the coordinate plane, what is the total length of the figure represented by point $\mathrm { P } ( x , y )$ satisfying the following two conditions? [4 points] (가) $x ^ { 2 } + y ^ { 2 } = 4$ (나) For any point $( 1 , a )$ on segment AB, the matrix $\left( \begin{array} { c c } x & y \\ 1 & a \end{array} \right)$ has an inverse matrix.
(1) $\frac { 1 } { 3 } \pi$
(2) $\frac { 1 } { 2 } \pi$
(3) $\pi$
(4) $\frac { 4 } { 3 } \pi$
(5) $\frac { 3 } { 2 } \pi$
csat-suneung 2006 Q27 4 marks Matrix Power Computation and Application View
For two matrices $A = \left( \begin{array} { l l } 1 & 1 \\ 0 & 1 \end{array} \right) , B = \left( \begin{array} { l l } 1 & 0 \\ 1 & 1 \end{array} \right)$, define sets $S , T$ as $$\begin{aligned} & S = \left\{ \binom { x } { y } \left\lvert \, \binom { x } { y } = A ^ { n } \binom { 1 } { 1 } \right. , n \text { is a natural number} \right\} \\ & T = \left\{ \binom { x } { y } \left\lvert \, \binom { x } { y } = B ^ { n } \binom { 1 } { 1 } \right. , n \text { is a natural number} \right\} \end{aligned}$$ Which of the following in are correct? [4 points] 〈Remarks〉 ㄱ. If $\binom { a } { b } \in S$, then $\binom { b } { a } \in T$. ㄴ. If $\binom { a } { b } \in S , \binom { c } { d } \in S$, then $\binom { a + c } { b + d } \in S$. ㄷ. If $\binom { a } { b } \in S , \binom { p } { q } \in T$, then the matrix $\left( \begin{array} { l l } a & p \\ b & q \end{array} \right)$ has an inverse matrix.
(1) ㄱ
(2) ㄱ, ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ
csat-suneung 2007 Q2 2 marks Linear System and Inverse Existence View
For two matrices $A = \left( \begin{array} { r r } - 1 & 0 \\ 0 & 1 \end{array} \right) , B = \left( \begin{array} { l l } 2 & 1 \\ 3 & 3 \end{array} \right)$, what is the sum of all components of the matrix $( A + B ) ^ { - 1 }$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
csat-suneung 2007 Q2 2 marks Linear System and Inverse Existence View
For two matrices $A = \left( \begin{array} { r r } - 1 & 0 \\ 0 & 1 \end{array} \right) , B = \left( \begin{array} { l l } 2 & 1 \\ 3 & 3 \end{array} \right)$, what is the sum of all components of the matrix $( A + B ) ^ { - 1 }$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
csat-suneung 2007 Q12 3 marks True/False or Multiple-Select Conceptual Reasoning View
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
csat-suneung 2007 Q12 3 marks True/False or Multiple-Select Conceptual Reasoning View
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
csat-suneung 2007 Q21 3 marks Linear System and Inverse Existence View
For a $2 \times 2$ square matrix $A$ satisfying $( A + E ) ^ { 2 } = A$ and a matrix $\binom { p } { q }$, $$\left( A + A ^ { - 1 } \right) \binom { p } { q } = \binom { 3 } { - 7 }$$ holds. Find the value of $p ^ { 2 } + q ^ { 2 }$. (Note: $E$ is the identity matrix.) [3 points]
csat-suneung 2007 Q30 4 marks Determinant and Rank Computation View
For a $2 \times 2$ square matrix $X = \left( \begin{array} { l l } a & b \\ c & d \end{array} \right)$, $$D ( X ) = a d - b c$$ is defined. For a $2 \times 2$ square matrix $A = \left( \begin{array} { l l } 1 & 1 \\ 0 & p \end{array} \right)$, $$D \left( A ^ { 2 } \right) = D ( 5 A )$$ Find the sum of all constants $p$ that satisfy this condition. [4 points]
csat-suneung 2008 Q2 2 marks Matrix Algebra and Product Properties View
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) $\left( \begin{array} { l l } - 3 & 1 \\ - 2 & 2 \end{array} \right)$
csat-suneung 2008 Q2 2 marks Matrix Algebra and Product Properties View
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) $\left( \begin{array} { l l } - 3 & 1 \\ - 2 & 2 \end{array} \right)$
csat-suneung 2008 Q5 3 marks Linear System and Inverse Existence View
For the matrix $A = \left( \begin{array} { c c } 2 n & - 7 \\ - 1 & n \end{array} \right)$, what is the natural number $n$ such that all components of the inverse matrix $A ^ { - 1 }$ are natural numbers? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
csat-suneung 2008 Q15 4 marks True/False or Multiple-Select Conceptual Reasoning View
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
csat-suneung 2008 Q15 4 marks True/False or Multiple-Select Conceptual Reasoning View
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
csat-suneung 2008 Q20 3 marks Matrix Power Computation and Application View
For the matrix $A = \left( \begin{array} { l l } 1 & 0 \\ 3 & 1 \end{array} \right)$ with $A ^ { 8 } = \left( \begin{array} { l l } 1 & 0 \\ a & 1 \end{array} \right)$, find the value of $a$. [3 points]

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