Let $\mathrm { A } = \left[ \begin{array} { l l l } 2 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{array} \right] , \mathrm { B } = \left[ \begin{array} { l l l } \mathrm { B } _ { 1 } & \mathrm {~B} _ { 2 } & \mathrm {~B} _ { 3 } \end{array} \right]$, where $\mathrm { B } _ { 1 } , \mathrm {~B} _ { 2 } , \mathrm {~B} _ { 3 }$ are column matrices, and $\mathrm { AB } _ { 1 } = \left[ \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right]$, $\mathrm { AB } _ { 2 } = \left[ \begin{array} { l } 2 \\ 3 \\ 0 \end{array} \right] , \mathrm { AB } _ { 3 } = \left[ \begin{array} { l } 3 \\ 2 \\ 1 \end{array} \right]$ If $\alpha = | \mathrm { B } |$ and $\beta$ is the sum of all the diagonal elements of B, then $\alpha ^ { 3 } + \beta ^ { 3 }$ is equal to

Consider the matrices : $A = \left[ \begin{array} { c c } 2 & - 5 \\ 3 & m \end{array} \right] , B = \left[ \begin{array} { l } 20 \\ m \end{array} \right]$ and $X = \left[ \begin{array} { l } x \\ y \end{array} \right]$. Let the set of all $m$, for which the system of equations $A X = B$ has a negative solution (i.e., $x < 0$ and $y < 0$ ), be the interval ( $a , b$ ). Then $8 \int _ { a } ^ { b } | A | d m$ is equal to $\_\_\_\_$

If the system of linear equations : $$x + y + 2z = 6$$ $$2x + 3y + \mathrm { a } z = \mathrm { a } + 1$$ $$- x - 3 y + \mathrm { b } z = 2 \mathrm {~b}$$ where $a , b \in \mathbf { R }$, has infinitely many solutions, then $7a + 3b$ is equal to :
(1) 16
(2) 12
(3) 22
(4) 9

The system of equations $$x + y + z = 6$$ $$x + 2 y + 5 z = 9$$ $$x + 5 y + \lambda z = \mu$$ has no solution if
(1) $\lambda = 15 , \mu \neq 17$
(2) $\lambda \neq 17 , \mu \neq 18$
(3) $\lambda = 17 , \mu \neq 18$
(4) $\lambda = 17 , \mu = 18$

Let $A = \left[ a _ { i j } \right]$ be $3 \times 3$ matrix such that $A \left[ \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right] = \left[ \begin{array} { l } 0 \\ 0 \\ 1 \end{array} \right] , A \left[ \begin{array} { l } 4 \\ 1 \\ 3 \end{array} \right] = \left[ \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right]$ and $A \left[ \begin{array} { l } 2 \\ 1 \\ 2 \end{array} \right] = \left[ \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right]$, then $a _ { 23 }$ equals :
(1) $- 1$
(2) 2
(3) 1
(4) 0

If the system of equations $$2x - y + z = 4$$ $$5x + \lambda y + 3z = 12$$ $$100x - 47y + \mu z = 212$$ has infinitely many solutions, then $\mu - 2\lambda$ is equal to
(1) 57
(2) 59
(3) 55
(4) 56

If the system of equations $$( \lambda - 1 ) x + ( \lambda - 4 ) y + \lambda z = 5$$ $$\lambda x + ( \lambda - 1 ) y + ( \lambda - 4 ) z = 7$$ $$( \lambda + 1 ) x + ( \lambda + 2 ) y - ( \lambda + 2 ) z = 9$$ has infinitely many solutions, then $\lambda ^ { 2 } + \lambda$ is equal to
(1) 6
(2) 10
(3) 20
(4) 12

If the system of equations $$x + 2y - 3z = 2$$ $$2x + \lambda y + 5z = 5$$ $$14x + 3y + \mu z = 33$$ has infinitely many solutions, then $\lambda + \mu$ is equal to :
(1) 13
(2) 10
(3) 12
(4) 11

Q84. Let $A$ be a $2 \times 2$ symmetric matrix such that $A \left[ \begin{array} { l } 1 \\ 1 \end{array} \right] = \left[ \begin{array} { l } 3 \\ 7 \end{array} \right]$ and the determinant of $A$ be 1 . If $A ^ { - 1 } = \alpha A + \beta I$, where $I$ is an identity matrix of order $2 \times 2$, then $\alpha + \beta$ equals $\_\_\_\_$

Q86. Let $\alpha \beta \gamma = 45 ; \alpha , \beta , \gamma \in \mathbb { R }$. If $x ( \alpha , 1,2 ) + y ( 1 , \beta , 2 ) + z ( 2,3 , \gamma ) = ( 0,0,0 )$ for some $x , y , z \in \mathbb { R } , x y z \neq 0$, then $6 \alpha + 4 \beta + \gamma$ is equal to $\_\_\_\_$

Q85. Consider the matrices : $A = \left[ \begin{array} { c c } 2 & - 5 \\ 3 & m \end{array} \right] , B = \left[ \begin{array} { l } 20 \\ m \end{array} \right]$ and $X = \left[ \begin{array} { l } x \\ y \end{array} \right]$. Let the set of all $m$, for which the system of equations $A X = B$ has a negative solution (i.e., $x < 0$ and $y < 0$ ), be the interval ( $a , b$ ). Then $8 \int _ { a } ^ { b } | A | d m$ is equal to $\_\_\_\_$

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) $\left(\begin{array}{llll} 1 & 3 & 2 & 7 \\ 0 & 1 & 1 & 2 \\ 0 & 1 & 0 & 1 \end{array}\right)$

Let $a , b$ be real numbers, and the system of equations $\left\{ \begin{array} { l } a x + 5 y + 12 z = 4 \\ x + a y + \frac { 8 } { 3 } z = 7 \\ 3 x + 8 y + a z = 1 \end{array} \right.$ has exactly one solution. After a series of Gaussian elimination operations, the original augmented matrix can be transformed to $\left[ \begin{array} { c c c | c } 1 & 2 & b & 7 \\ 0 & b & 5 & - 5 \\ 0 & 0 & b & 0 \end{array} \right]$ . Then $a = ( 14 - 1 ) , b = \frac { ( 14 - 2 ) } { ( 14 - 3 ) }$ . (Express as a fraction in lowest terms)

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).

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

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

$$\begin{aligned} & A = \left[ \begin{array} { l l } 2 & 3 \\ 1 & 2 \end{array} \right] \\ & B = \left[ \begin{array} { l l } 1 & 2 \\ 0 & 5 \end{array} \right] \end{aligned}$$
With the matrix notation
$$( 2 A - B ) \cdot \left[ \begin{array} { l } x \\ y \end{array} \right] = \left[ \begin{array} { l } 1 \\ 0 \end{array} \right]$$
Which of the following is the system of linear equations?
A) $\begin{aligned} & x - 4 y = 0 \\ & 2 x - y = 1 \end{aligned}$
B) $\begin{aligned} & x + 2 y = 0 \\ & 2 x - 3 y = 1 \end{aligned}$
C) $\begin{aligned} & 2 x + y = 1 \\ & x - y = 0 \end{aligned}$
D) $\begin{aligned} & 3 x - 2 y = 1 \\ & 2 x + y = 0 \end{aligned}$
E) $\begin{aligned} & 3 x + 4 y = 1 \\ & 2 x - y = 0 \end{aligned}$

Let I be the $2 \times 2$ identity matrix and
$$A = \left[ \begin{array} { l l } 4 & 5 \\ 1 & 3 \end{array} \right]$$
Accordingly, which of the following is $( \mathbf { A } - \mathbf { I } ) ^ { - \mathbf { 1 } }$ equal to?
A) $\left[ \begin{array} { r r } 2 & - 5 \\ - 1 & 3 \end{array} \right]$
B) $\left[ \begin{array} { r r } 1 & - 4 \\ - 2 & 3 \end{array} \right]$
C) $\left[ \begin{array} { r r } 0 & - 1 \\ - 1 & 4 \end{array} \right]$
D) $\left[ \begin{array} { l l } - 2 & 5 \\ - 1 & 0 \end{array} \right]$
E) $\left[ \begin{array} { l l } 2 & - 5 \\ 0 & - 3 \end{array} \right]$

Let $A$ and $B$ be $2 \times 1$ matrices and $t$ be a variable such that for all $x$ and $y$ values satisfying
$$x - y = 3$$
we have
$$\left[ \begin{array} { l } x \\ y \end{array} \right] = t A + B$$
Accordingly, which of the following could matrices A and B be, respectively?
A) $\left[ \begin{array} { l } 1 \\ 0 \end{array} \right] , \left[ \begin{array} { l } 3 \\ 0 \end{array} \right]$
B) $\left[ \begin{array} { l } 0 \\ 1 \end{array} \right] , \left[ \begin{array} { l } 3 \\ 0 \end{array} \right]$
C) $\left[ \begin{array} { l } 1 \\ 1 \end{array} \right] , \left[ \begin{array} { l } 3 \\ 1 \end{array} \right]$
D) $\left[ \begin{array} { l } 1 \\ 1 \end{array} \right] , \left[ \begin{array} { l } 3 \\ 0 \end{array} \right]$
E) $\left[ \begin{array} { l } 1 \\ 1 \end{array} \right] , \left[ \begin{array} { r } 3 \\ - 3 \end{array} \right]$

The inverse of matrix A is $A ^ { - 1 } = \left[ \begin{array} { l l } 1 & 0 \\ 2 & 1 \end{array} \right]$. Given that
$$A \cdot \left[ \begin{array} { l } 1 \\ a \end{array} \right] = \left[ \begin{array} { l } b \\ 4 \end{array} \right]$$
what is the sum $\mathrm { a } + \mathrm { b }$?
A) 5
B) 7
C) 8
D) 9
E) 11

$$3 x - y = 2$$ $$5 x + 2 y = 3$$
The matrix representation of the linear equation system is
$$A \cdot \left[ \begin{array} { l } x \\ y \end{array} \right] = \left[ \begin{array} { l } 2 \\ 3 \end{array} \right]$$
Given that
$$A \cdot \left[ \begin{array} { l } 1 \\ 2 \end{array} \right] = \left[ \begin{array} { l } a \\ b \end{array} \right]$$
what is the sum $\mathbf { a + b }$?
A) 4
B) 6
C) 8
D) 10
E) 12