LFM Pure > Matrices

Let $A$ be a matrix of size $3 \times 4$ such that its first two rows are $(1,1,1,1)$ and $(1,2,3,4)$, and with no zeros in the third row. In each of the following sections, provide an example of matrix $A$ that satisfies the requested condition, justifying it appropriately:\ a) (0.5 points) The third row of $A$ is a linear combination of the first two.\ b) (0.5 points) The three rows of $A$ are linearly independent.\ c) (0.5 points) $A$ is the augmented matrix of a compatible determined system.\ d) (0.5 points) $A$ is the augmented matrix of a compatible indeterminate system.\ e) (0.5 points) $A$ is the augmented matrix of an incompatible system.

spain-selectividad 2022 Q5 2.5 marks Determinant and Rank Computation

Consider the real matrices
$$A = \left( \begin{array} { c c c } 1 & - 1 & k \\ k & 1 & - 1 \end{array} \right) , \quad B = \left( \begin{array} { c c } 1 & 1 \\ 1 & - 1 \\ 1 & 0 \end{array} \right)$$
a) (1 point) Calculate for which values of the parameter $k$ the matrix AB has an inverse. Calculate the inverse matrix of AB for $k = 1$. b) (1 point) Calculate BA and discuss its rank as a function of the value of the real parameter $k$. c) ( 0.5 points) In the case $k = 1$, write an inconsistent system of three linear equations with three unknowns whose coefficient matrix is BA.

spain-selectividad 2023 QA.1 2 marks Determinant and Rank Computation

Given the matrices $A = \left( \begin{array} { c c c } 2 & 1 & 0 \\ - 1 & 0 & 2 \end{array} \right)$ and $B = \left( \begin{array} { l l } b & 0 \\ 1 & b \end{array} \right)$, find:\ a) ( 0.5 points) Calculate the determinant of $A ^ { t } A$.\ b) ( 0.5 points) Calculate the rank of $B A$ as a function of $b$.\ c) (0.75 points) Calculate $B ^ { - 1 }$ for $b = 2$.\ d) ( 0.75 points) For $b = 1$, calculate $B ^ { 5 }$.

spain-selectividad 2024 QB.1 2.5 marks Determinant and Rank Computation

Consider the real matrices $A = \left( \begin{array} { c c c } m & 1 & 1 \\ 0 & m & 3 \end{array} \right)$ and $B = \left( \begin{array} { c c } 1 & m \\ 0 & m \\ 0 & 1 \end{array} \right)$. It is requested:
a) ( 0.75 points) Study whether there exists some value of $m$ for which the matrix $B A$ has an inverse.
b) ( 0.75 points) Study the rank of the matrix $A B$ as a function of the parameter $m$.
c) (1 point) For $m = 1$, discuss the system $\left( A ^ { t } A \right) \left( \begin{array} { l } x \\ y \\ z \end{array} \right) = \left( \begin{array} { c } a \\ a \\ a ^ { 2 } \end{array} \right)$ according to the values of $a$.

spain-selectividad 2025 Q1.1 2.5 marks Determinant and Rank Computation

Let $\lambda$ be a real number and consider the matrices $A = \left( \begin{array} { c c c } \lambda & 1 & \lambda \\ 0 & \lambda & - 1 \end{array} \right)$ and $B = \left( \begin{array} { c c } 1 & \lambda \\ 0 & - 1 \\ 1 & - \lambda \end{array} \right)$. It is requested: a) ( 0.5 points) Determine whether there exists some value of $\lambda$ for which the matrix $AB$ does not have an inverse. b) (1 point) Study the rank of the matrix $BA$ as a function of the parameter $\lambda$. c) (1 point) For $\lambda = 1$, discuss the system $\left( A ^ { t } A \right) \left( \begin{array} { c } x \\ y \\ z \end{array} \right) = \left( \begin{array} { c } a ^ { 2 } \\ a ^ { 2 } \\ 2 a \end{array} \right)$ according to the values of $a$.

taiwan-gsat 2007 Q8 Linear System and Inverse Existence

8. Which of the following matrices can be transformed into $\left(\begin{array}{llll} 1 & 2 & 3 & 7 \\ 0 & 1 & 1 & 2 \\ 0 & 0 & 1 & 1 \end{array}\right)$ through a series of row operations?
(1) $\left(\begin{array}{llll} 1 & 2 & 3 & 7 \\ 0 & 1 & 1 & 2 \\ 0 & 2 & 3 & 5 \end{array}\right)$
(2) $\left(\begin{array}{cccc} -1 & 3 & -1 & 0 \\ -1 & 1 & 1 & 0 \\ 3 & 1 & -7 & 0 \end{array}\right)$
(3) $\left(\begin{array}{cccc} 1 & 1 & 2 & 5 \\ 1 & -1 & 1 & 2 \\ 1 & 1 & 2 & 5 \end{array}\right)$
(4) $\left(\begin{array}{cccc} 2 & 1 & 3 & 6 \\ -1 & 1 & 1 & 0 \\ -2 & 2 & 2 & 1 \end{array}\right)$
(5) $\left(\begin{array}{llll} 1 & 3 & 2 & 7 \\ 0 & 1 & 1 & 2 \\ 0 & 1 & 0 & 1 \end{array}\right)$

taiwan-gsat 2010 Q2 Determinant and Rank Computation

2. Given that $a, b$ are integers and the determinant $\left| \begin{array} { c c } 5 & a \\ b & 7 \end{array} \right| = 4$, what is the absolute value $|a + b|$?
(1) 16
(2) 31
(3) 32
(4) 39
(5) Insufficient information to determine

taiwan-gsat 2020 Q1 6 marks Matrix Power Computation and Application

Which of the following matrices is equal to $\left[ \begin{array} { c c } - 1 & 0 \\ 1 & - 1 \end{array} \right] ^ { 5 }$ ?
(1) $\left[ \begin{array} { c c } - 1 & 0 \\ - 5 & - 1 \end{array} \right]$
(2) $\left[ \begin{array} { c c } 1 & 0 \\ - 5 & 1 \end{array} \right]$
(3) $\left[ \begin{array} { c c } - 1 & 5 \\ 0 & - 1 \end{array} \right]$
(4) $\left[ \begin{array} { l l } 1 & 0 \\ 5 & 1 \end{array} \right]$
(5) $\left[ \begin{array} { c c } - 1 & 0 \\ 5 & - 1 \end{array} \right]$

taiwan-gsat 2020 Q8 8 marks True/False or Multiple-Select Conceptual Reasoning

Let a $2 \times 2$ real matrix $A$ represent a reflection transformation of the coordinate plane and satisfy $A^{3} = \left[\begin{array}{cc} 0 & -1 \\ -1 & 0 \end{array}\right]$; let a $2 \times 2$ real matrix $B$ represent a rotation transformation (centered at the origin) of the coordinate plane and satisfy $B^{3} = \left[\begin{array}{cc} -1 & 0 \\ 0 & -1 \end{array}\right]$. Select the correct options.
(1) There are exactly three possible matrices for $A$
(2) There are exactly three possible matrices for $B$
(3) $AB = BA$
(4) The $2 \times 2$ matrix $AB$ represents a rotation transformation of the coordinate plane
(5) $BABA = \left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]$

taiwan-gsat 2021 QB 8 marks Matrix Power Computation and Application

Let matrix $A = \left[ \begin{array} { c c } 1 & - 2 \\ 0 & 1 \end{array} \right] \left[ \begin{array} { l l } 1 & 0 \\ 0 & 6 \end{array} \right] \left[ \begin{array} { c c } 1 & - 2 \\ 0 & 1 \end{array} \right] ^ { - 1 } , B = \left[ \begin{array} { c c } 1 & - 2 \\ 0 & 1 \end{array} \right] \left[ \begin{array} { l l } 6 & 0 \\ 0 & 1 \end{array} \right] \left[ \begin{array} { c c } 1 & - 2 \\ 0 & 1 \end{array} \right] ^ { - 1 }$, where $\left[ \begin{array} { c c } 1 & - 2 \\ 0 & 1 \end{array} \right] ^ { - 1 }$ is the inverse matrix of $\left[ \begin{array} { c c } 1 & - 2 \\ 0 & 1 \end{array} \right]$. If $A + B = \left[ \begin{array} { l l } a & b \\ c & d \end{array} \right]$, then $a + b + c + d =$ (10)(11).

taiwan-gsat 2021 Q1 5 marks Matrix Power Computation and Application

Let $A = \left[ \begin{array} { l l } 1 & 2 \\ 0 & 3 \end{array} \right]$. If $A ^ { 4 } = \left[ \begin{array} { l l } a & b \\ c & d \end{array} \right]$, what is the value of $a + b + c + d$?
(1) 158
(2) 162
(3) 166
(4) 170
(5) 174

taiwan-gsat 2022 Q5 5 marks Matrix Power Computation and Application

Let matrix $A = \left[\begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array}\right]$. If $A^{7} - 3A = \left[\begin{array}{ll} a & b \\ c & d \end{array}\right]$, then which of the following is the value of $a + b + c + d$?
(1) $-8$
(2) $-5$
(3) $5$
(4) $8$
(5) $10$

taiwan-gsat 2023 Q2 5 marks Matrix Algebra and Product Properties

Consider a real $2 \times 2$ matrix $\left[ \begin{array} { l l } a & b \\ c & d \end{array} \right]$. If $\left[ \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right] \left[ \begin{array} { l l } a & b \\ c & d \end{array} \right] \left[ \begin{array} { c c } 1 & 0 \\ 0 & - 2 \end{array} \right] = \left[ \begin{array} { c c } 3 & - 4 \\ - 9 & - 7 \end{array} \right]$, what is the value of $c - 2b$?
(1) $- 11$ (2) $- 4$ (3) $1$ (4) $10$ (5) $11$

taiwan-gsat 2023 Q6 8 marks True/False or Multiple-Select Conceptual Reasoning

Let $a , b , c , d , r , s , t$ all be real numbers. It is known that three non-zero vectors $\vec { u } = ( a , b , 0 )$, $\vec { v } = ( c , d , 0 )$, and $\vec { w } = ( r , s , t )$ in coordinate space satisfy the dot products $\vec { w } \cdot \vec { u } = \vec { w } \cdot \vec { v } = 0$. Consider the $3 \times 3$ matrix $A = \left[ \begin{array} { l l l } a & b & 0 \\ c & d & 0 \\ r & s & t \end{array} \right]$. Select the correct options.
(1) If $\vec { u } \cdot \vec { v } = 0$, then the determinant $\left| \begin{array} { l l } a & b \\ c & d \end{array} \right| \neq 0$
(2) If $t \neq 0$, then the determinant $\left| \begin{array} { l l } a & b \\ c & d \end{array} \right| \neq 0$
(3) If there exists a vector $\overrightarrow { w ^ { \prime } }$ satisfying $\overrightarrow { w ^ { \prime } } \cdot \vec { u } = \overrightarrow { w ^ { \prime } } \cdot \vec { v } = 0$ and cross product $\overrightarrow { w ^ { \prime } } \times \vec { w } \neq \overrightarrow { 0 }$, then the determinant $\left| \begin{array} { l l } a & b \\ c & d \end{array} \right| \neq 0$
(4) If for any three real numbers $e , f , g$, the vector $( e , f , g )$ can be expressed as a linear combination of $\vec { u } , \vec { v } , \vec { w }$, then the determinant $\left| \begin{array} { l l } a & b \\ c & d \end{array} \right| \neq 0$
(5) If the determinant $\left| \begin{array} { l l } a & b \\ c & d \end{array} \right| \neq 0$, then the determinant of $A$ is not equal to 0

taiwan-gsat 2023 Q11 5 marks True/False or Multiple-Select Conceptual Reasoning

On the coordinate plane, let $A$ and $B$ denote the rotation matrices for clockwise and counterclockwise rotation by $90^{\circ}$ about the origin respectively. Let $C$ and $D$ denote the reflection matrices with reflection axes $x = y$ and $x = -y$ respectively. Select the correct options.
(1) $A$ and $C$ map the point $(1,0)$ to the same point
(2) $A = -B$
(3) $C = D^{-1}$
(4) $AB = CD$
(5) $AC = BD$

taiwan-gsat 2024 Q13 5 marks Linear System and Inverse Existence

Given that $a , b , c , d$ are real numbers, and $\left[ \begin{array} { l l } 1 & - 1 \\ 3 & - 2 \end{array} \right] \left[ \begin{array} { l } a \\ b \end{array} \right] = \left[ \begin{array} { l } 1 \\ 0 \end{array} \right]$. If $\left[ \begin{array} { l l } 1 & - 1 \\ 3 & - 2 \end{array} \right] \left[ \begin{array} { l } 2 a + 1 \\ 2 b + 1 \end{array} \right] = \left[ \begin{array} { l } c \\ d \end{array} \right]$, then the value of $c - 3 d$ is (13-1)(13-2).

taiwan-gsat 2025 Q2 5 marks Linear System and Inverse Existence

Let $A$ be a $3 \times 2$ matrix such that $A \left[ \begin{array} { c c } 1 & 0 \\ - 1 & 1 \end{array} \right] = \left[ \begin{array} { c c } 4 & - 6 \\ - 2 & 1 \\ 3 & 5 \end{array} \right]$ . If $A \left[ \begin{array} { l } 1 \\ 0 \end{array} \right] = \left[ \begin{array} { l } a \\ b \\ c \end{array} \right]$ , what is the value of $a + b + c$?
(1) 0
(2) 2
(3) 4
(4) 5
(5) 8

taiwan-gsat 2025 Q7 8 marks True/False or Multiple-Select Conceptual Reasoning

Let the second-order matrices $A = \left[ \begin{array} { l l } 1 & 0 \\ 1 & 0 \end{array} \right], B = \left[ \begin{array} { l l } 0 & 1 \\ 0 & 1 \end{array} \right]$. Select the correct options.
(1) $A ^ { 2 } = A$
(2) $A + B = B + A$
(3) $A B = B A$
(4) $( A - B ) ^ { 2 } = A ^ { 2 } - 2 A B + B ^ { 2 }$
(5) $( A + B ) ^ { 2 } = 2 ( A + B )$

taiwan-gsat 2025 Q9 6 marks Matrix Entry and Coefficient Identities

Let $A$ be the rotation matrix that rotates counterclockwise by angle $\theta$ about the origin, and let $B$ be the reflection matrix with the $x$-axis as the axis of reflection (axis of symmetry). Let $A = \left[\begin{array}{ll} a_{1} & a_{2} \\ a_{3} & a_{4} \end{array}\right]$ and $BA = \left[\begin{array}{ll} c_{1} & c_{2} \\ c_{3} & c_{4} \end{array}\right]$.
Given that $a_{1} + a_{2} + a_{3} + a_{4} = 2(c_{1} + c_{2} + c_{3} + c_{4})$, then $\tan\theta =$ . (Express as a fraction in lowest terms)

todai-math 2022 QI Matrix Algebraic Properties and Abstract Reasoning

Real symmetric matrices $\boldsymbol{A}$ and $\boldsymbol{B}$ are defined as follows:
$$\begin{aligned} & \boldsymbol{A} = \left( \begin{array}{ccc} 7 & -2 & 1 \\ -2 & 10 & -2 \\ 1 & -2 & 7 \end{array} \right), \\ & \boldsymbol{B} = \left( \begin{array}{ccc} 5 & -1 & -1 \\ -1 & 5 & -1 \\ -1 & -1 & 5 \end{array} \right). \end{aligned}$$
1. Obtain $\boldsymbol{AB}$.
Matrices $\boldsymbol{A}$ and $\boldsymbol{B}$ defined as Equations (1) and (2) satisfy $\boldsymbol{AB} = \boldsymbol{BA}$.
2. In general, two real symmetric matrices that are commutative for multiplication are simultaneously diagonalizable. Prove this for the case where all the eigenvalues are mutually different.
3. Suppose a three-dimensional real vector $\boldsymbol{v}$ whose norm is 1 is an eigenvector of $\boldsymbol{A}$ in Equation (1) corresponding to an eigenvalue $a$ as well as an eigenvector of $\boldsymbol{B}$ in Equation (2) corresponding to an eigenvalue $b$. That is, $\boldsymbol{Av} = a\boldsymbol{v}$, $\boldsymbol{Bv} = b\boldsymbol{v}$, and $\|\boldsymbol{v}\| = 1$. Obtain all the sets of $(\boldsymbol{v}, a, b)$.

todai-math 2022 QII Matrix Algebraic Properties and Abstract Reasoning

Answer the following questions concerning the curved surface given by Equation (3) in the Cartesian coordinate system $xyz$. Note that $\boldsymbol{m}^{\mathrm{T}}$ indicates transpose of $\boldsymbol{m}$.
$$f(x, y, z) = 2\left(x^{2} + y^{2} + z^{2}\right) + 4yz + \frac{z - y}{\sqrt{2}} = 0 \tag{3}$$
1. When the function $f(x, y, z)$ is expressed in the following form, derive the real symmetric matrix $\boldsymbol{A}$ of order 3 and the vector $\boldsymbol{b} = \left(\begin{array}{l} b_{1} \\ b_{2} \\ b_{3} \end{array}\right)$:
$$f(x, y, z) = \left(\begin{array}{lll} x & y & z \end{array}\right) \boldsymbol{A} \left(\begin{array}{l} x \\ y \\ z \end{array}\right) + 2\boldsymbol{b}^{\mathrm{T}} \left(\begin{array}{l} x \\ y \\ z \end{array}\right)$$
2. Suppose that the matrix $\boldsymbol{A}$ derived in Question II.1 is diagonalized as $\boldsymbol{A} = \boldsymbol{P}^{\mathrm{T}}\boldsymbol{D}\boldsymbol{P}$ using an orthogonal matrix $\boldsymbol{P}$ of order 3 and a diagonal matrix $\boldsymbol{D}$, which is given by Equation (5):
$$\boldsymbol{D} = \left(\begin{array}{ccc} d_{1} & 0 & 0 \\ 0 & d_{2} & 0 \\ 0 & 0 & d_{3} \end{array}\right) \tag{5}$$
Obtain a set of $\boldsymbol{P}$ and $\boldsymbol{D}$ satisfying $d_{1} \geq d_{2} \geq d_{3}$.
3. Express the function $f$ using $X$, $Y$, and $Z$, obtained by applying the coordinate transformation defined by $\left(\begin{array}{l} X \\ Y \\ Z \end{array}\right) = \boldsymbol{P} \left(\begin{array}{l} x \\ y \\ z \end{array}\right)$, using $\boldsymbol{P}$ derived in Question II.2.
4. Consider a region surrounded by the curved surface given by Equation (3) and a plane defined by $y - z - \sqrt{2} = 0$. Obtain the volume of this region.

turkey-yks 2010 Q33 Direct Determinant Computation

$$\left|\begin{array}{rrr} 2 & -3 & 2 \\ 1 & 2 & 0 \\ 2 & 3 & 0 \end{array}\right|$$
What is the value of this determinant?
A) $-1$
B) $-2$
C) $-3$
D) $-4$
E) $-6$

turkey-yks 2010 Q34 Matrix Algebra and Product Properties

$$A = \begin{bmatrix} 2 & 4 \\ 1 & 3 \end{bmatrix}$$
Given that $A^{t}$ is the transpose of the matrix and $A^{-1}$ is its inverse matrix, which of the following is the product $A^{t} \cdot A^{-1}$?
A) $\begin{bmatrix} \frac{5}{2} & -3 \\ \frac{9}{2} & -5 \end{bmatrix}$
B) $\begin{bmatrix} \frac{3}{2} & -2 \\ 1 & 3 \end{bmatrix}$
C) $\begin{bmatrix} -2 & \frac{-9}{2} \\ 3 & \frac{5}{2} \end{bmatrix}$
D) $\begin{bmatrix} \frac{9}{2} & 3 \\ \frac{-5}{2} & -1 \end{bmatrix}$
E) $\begin{bmatrix} -3 & -1 \\ \frac{5}{2} & -2 \end{bmatrix}$

turkey-yks 2012 Q35 Matrix Algebra and Product Properties

Let a, b and c be positive real numbers,
$$\left[ \begin{array} { l l } a & b \\ 0 & c \end{array} \right] \cdot \left[ \begin{array} { l l } a & b \\ 0 & c \end{array} \right] = \left[ \begin{array} { l l } 1 & 2 \\ 0 & 4 \end{array} \right]$$
The matrix equation is given. Accordingly, what is the sum $a + b + c$?
A) $\frac { 11 } { 3 }$
B) $\frac { 7 } { 4 }$
C) 4
D) 5
E) 6

turkey-yks 2012 Q36 Linear System and Inverse Existence

For a matrix A with multiplicative inverse $A^{-1}$,
$$\left[ \begin{array} { l l } 2 & 1 \end{array} \right] \cdot \left[ \begin{array} { l l } 1 & 0 \\ 3 & 1 \end{array} \right] ^ { - 1 } \cdot \left[ \begin{array} { l } 1 \\ 4 \end{array} \right] = [ a ]$$
In the matrix equation, what is a?
A) 1
B) 2
C) 3
D) 4
E) 5