If $A = \left[ \begin{array} { l l } 2 & 3 \\ 3 & 5 \end{array} \right]$, then the value of $\left| \vec { A } ^ { 2025 } - 3 A ^ { 2024 } + A ^ { 2023 } \right|$ is
Given the matrices $A = \left( \begin{array} { c c c c } 1 & 3 & 4 & 1 \\ 1 & a & 2 & 2 - a \\ - 1 & 2 & a & a - 2 \end{array} \right)$ and $M = \left( \begin{array} { c c c } 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{array} \right)$; find: a) ( 1.5 points) Study the rank of A as a function of the real parameter a. b) ( 1 point) Calculate, if possible, the inverse of the matrix AM for the case a $= 0$.
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.
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.
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 $B ^ { 5 }$.
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 $a$.
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 $a$.
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
Assume vectors $\boldsymbol{a}_1, \boldsymbol{a}_2, \ldots, \boldsymbol{a}_m$ are linearly independent in a vector space $V$, where $m$ is an integer greater than or equal to 3. Obtain the condition that $m$ must satisfy in order for $\boldsymbol{a}_1 + \boldsymbol{a}_2,\ \boldsymbol{a}_2 + \boldsymbol{a}_3,\ \ldots,\ \boldsymbol{a}_{m-1} + \boldsymbol{a}_m$ and $\boldsymbol{a}_m + \boldsymbol{a}_1$ to be linearly independent.
Answer the following questions. (1) The function $f ( x , y )$ with real variables $x , y$ is defined as follows: $$f ( x , y ) = \left| \begin{array} { c c c }
1 & x _ { 1 } & y _ { 1 } \\
1 & x _ { 2 } & y _ { 2 } \\
1 & x & y
\end{array} \right|$$ Show that the set of solutions of the equation $f ( x , y ) = 0$ is a line passing through two points $\left( x _ { 1 } , y _ { 1 } \right) , \left( x _ { 2 } , y _ { 2 } \right)$ on the $x y$ plane, where $x _ { 1 } \neq x _ { 2 }$. (2) Find the value of the determinant $\left| \begin{array} { c c c } 1 & x _ { 1 } & x _ { 1 } ^ { 2 } \\ 1 & x _ { 2 } & x _ { 2 } ^ { 2 } \\ 1 & x _ { 3 } & x _ { 3 } ^ { 2 } \end{array} \right|$ in factored form. (3) Show that there is a unique curve $y = a _ { 0 } + a _ { 1 } x + a _ { 2 } x ^ { 2 }$ passing through three points $\left( x _ { 1 } , y _ { 1 } \right) , \left( x _ { 2 } , y _ { 2 } \right) , \left( x _ { 3 } , y _ { 3 } \right)$ on the $x y$ plane, where $a _ { 0 } , a _ { 1 } , a _ { 2 }$ are constants and $x _ { 1 } , x _ { 2 } , x _ { 3 }$ are all distinct. (4) The curve in (3) can be represented in the form $y = c _ { 1 } y _ { 1 } + c _ { 2 } y _ { 2 } + c _ { 3 } y _ { 3 }$, where each of $c _ { 1 } , c _ { 2 } , c _ { 3 }$ does not depend on $y _ { 1 } , y _ { 2 } , y _ { 3 }$. Find $c _ { 1 } , c _ { 2 } , c _ { 3 }$. (5) Let us represent a curve $y = a _ { 0 } + a _ { 1 } x + a _ { 2 } x ^ { 2 } + a _ { 3 } x ^ { 3 } + a _ { 4 } x ^ { 4 }$ passing through five points $\left( x _ { 1 } , y _ { 1 } \right) , \ldots , \left( x _ { 5 } , y _ { 5 } \right)$ on the $x y$ plane in the form $y = c _ { 1 } y _ { 1 } + \cdots + c _ { 5 } y _ { 5 }$, where each of $c _ { 1 } , \ldots , c _ { 5 }$ does not depend on $y _ { 1 } , \ldots , y _ { 5 }$, and $x _ { 1 } , \ldots , x _ { 5 }$ are all distinct. Find $c _ { 1 }$.
Let $\mathbb { R } ^ { 3 }$ be the set of the three-dimensional real column vectors and $\mathbb { R } ^ { 3 \times 3 }$ be the set of the three-by-three real matrices. Let $n _ { 1 } , n _ { 2 }$, and $n _ { 3 } \in \mathbb { R } ^ { 3 }$ be linearly independent unit-length vectors and $n _ { 4 } \in \mathbb { R } ^ { 3 }$ be a unit-length vector not parallel to $n _ { 1 } , n _ { 2 }$, or $n _ { 3 }$. Let A and B be square matrices defined as $$\mathbf { A } = \left( \begin{array} { l }
n _ { 1 } ^ { \mathrm { T } } - n _ { 2 } ^ { \mathrm { T } } \\
n _ { 2 } ^ { \mathrm { T } } - n _ { 3 } ^ { \mathrm { T } } \\
n _ { 3 } ^ { \mathrm { T } } - n _ { 4 } ^ { \mathrm { T } }
\end{array} \right) , \quad \mathbf { B } = \sum _ { i = 1 } ^ { 4 } n _ { i } n _ { i } ^ { \mathrm { T } }$$ Here, $\mathrm { X } ^ { \mathrm { T } }$ and $\boldsymbol { x } ^ { \mathrm { T } }$ denote the transpose of a matrix X and a vector $\boldsymbol { x }$, respectively. Answer the following questions. (1) Find the condition for $n _ { 4 }$ such that the rank of $\mathbf { A }$ is three. (2) In the three-dimensional Euclidean space $\mathbb { R } ^ { 3 }$, consider four planes $\Pi _ { i } = \{ x \in \left. \mathbb { R } ^ { 3 } \mid n _ { i } ^ { \mathrm { T } } \boldsymbol { x } - d _ { i } = 0 \right\}$ ( $d _ { i }$ is a real number, and $i = 1,2,3,4$ ) that satisfy the following three conditions: (i) the rank of A is three, (ii) $\Omega = \left\{ x \in \mathbb { R } ^ { 3 } \mid n _ { i } ^ { \mathrm { T } } x - d _ { i } \geq 0 , i = 1,2,3,4 \right\}$ is not the empty set, and (iii) there exists a sphere $\mathrm { C } ( \subset \Omega )$ to which $\Pi _ { i } ( i = 1,2,3,4 )$ are tangent. The position vector of the center of C is represented by $\mathbf { A } ^ { -1 } \boldsymbol { u }$ using a vector $\boldsymbol { u } \in \mathbb { R } ^ { 3 }$. Express $\boldsymbol { u }$ using $d _ { i } ( i = 1,2,3,4 )$. (3) Show that B is a positive definite symmetric matrix. (4) Consider the point P from which the sum of squared distances to four planes $\{ x \in \left. \mathbb { R } ^ { 3 } \mid n _ { i } ^ { \mathrm { T } } x - d _ { i } = 0 \right\}$ ( $d _ { i }$ is a real number, and $i = 1,2,3,4$ ) is minimized. The position vector of P is represented by $\mathrm { B } ^ { -1 } v$ using a vector $v \in \mathbb { R } ^ { 3 }$. Express $v$ using $n _ { i }$ and $d _ { i } ( i = 1,2,3,4 )$. (5) Let $l _ { i }$ be a straight line through a point $Q _ { i }$, the position vector of which is $x _ { i } \in \mathbb { R } ^ { 3 }$, parallel to $n _ { i } ( i = 1,2,3 )$ in $\mathbb { R } ^ { 3 }$. Let $\mathrm { R } _ { i }$ be the orthogonal projection of an arbitrary point $R$, the position vector of which is $y \in \mathbb { R } ^ { 3 }$, onto $l _ { i }$. The position vector of $R _ { i }$ is represented by $y - \mathrm { W } _ { i } \left( y - x _ { i } \right)$ using a matrix $\mathrm { W } _ { i } \in \mathbb { R } ^ { 3 \times 3 }$. The identity matrix is denoted by $I \in \mathbb { R } ^ { 3 \times 3 }$. (a) Express $\mathrm { W } _ { i }$ using $n _ { i }$ and I. (b) Show that $\mathrm { W } _ { i } ^ { \mathrm { T } } \mathrm { W } _ { i } = \mathrm { W } _ { i }$. (c) Consider a plane $\Sigma = \left\{ \boldsymbol { x } \in \mathbb { R } ^ { 3 } \mid \boldsymbol { a } ^ { \mathrm { T } } \boldsymbol { x } = b \right\} \left( \boldsymbol { a } \in \mathbb { R } ^ { 3 } \right.$ is a non-zero vector, and $b$ is a real number). Let $\mathrm { S } \in \Sigma$ be the point from which the sum of squared distances to $l _ { 1 } , l _ { 2 }$, and $l _ { 3 }$ is minimized. When $n _ { 1 } , n _ { 2 }$, and $n _ { 3 }$ are orthogonal to each other, the position vector of $S$ is represented by $$\left( \mathrm { I } - \frac { a a ^ { \mathrm { T } } } { a ^ { \mathrm { T } } a } \right) w + \frac { a b } { a ^ { \mathrm { T } } a }$$ using a vector $\boldsymbol { w } \in \mathbb { R } ^ { 3 }$ which is independent of $a$ and $b$. Express $\boldsymbol { w }$ using $\mathbf { W } _ { i }$ and $x _ { i } ( i = 1,2,3 )$.
$$A = \left[ \begin{array} { l l }
3 & 2 \\
0 & 1
\end{array} \right]$$ Given this, what is the value of the determinant $\left| A - A ^ { \top } \right|$? A) 3 B) 4 C) 5 D) 6 E) 9
$$\left[ \begin{array} { l l }
3 & 2 \\
1 & 0
\end{array} \right] \cdot \mathrm { A } = \left[ \begin{array} { c c }
- 2 & 4 \\
1 & 5
\end{array} \right]$$ What is the determinant of matrix A that satisfies this equation? A) 4 B) 5 C) 6 D) 7 E) 8
Let A be a $2 \times 2$ matrix and $I$ be the $2 \times 2$ identity matrix such that $$A ^ { 2 } = \left[ \begin{array} { l l }
2 & 1 \\
1 & 5
\end{array} \right]$$ What is the value of the determinant $| ( \mathbf { A } - \mathbf { I } ) ( \mathbf { A } + \mathbf { I } ) |$? A) 2 B) 3 C) 4 D) 5 E) 6
$$A = \left[ \begin{array} { r r } 1 & 0 \\ - 1 & 3 \end{array} \right], \quad B = \left[ \begin{array} { r r } - 1 & 1 \\ 0 & m \end{array} \right]$$ The matrices satisfy the equality $$\operatorname { det } ( A + B ) = \operatorname { det } ( A ) + \operatorname { det } ( B )$$ Accordingly, what is m? A) - 3 B) - 1 C) 0 D) 2 E) 4