Let the system of linear equations $x + y + a z = 2$ $3 x + y + z = 4$ $x + 2 z = 1$ have a unique solution $\left( x ^ { * } , y ^ { * } , z ^ { * } \right)$. If $\left( \left( a , x ^ { * } \right) , \left( y ^ { * } , \alpha \right) \right.$ and $\left( x ^ { * } , - y ^ { * } \right)$ are collinear points, then the sum of absolute values of all possible values of $\alpha$ is:
(1) 4
(2) 3
(3) 2
(4) 1

The number of values of $\alpha$ for which the system of equations $x + y + z = \alpha$ $\alpha x + 2 \alpha y + 3 z = - 1$ $x + 3 \alpha y + 5 z = 4$ is inconsistent, is
(1) 0
(2) 1
(3) 2
(4) 3

Let $A$ be a $3 \times 3$ real matrix such that $A \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}$; $A \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}$ and $A \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}$. If $X = \begin{pmatrix} x _ { 1 } \\ x _ { 2 } \\ x _ { 3 } \end{pmatrix}$ and $I$ is an identity matrix of order 3, then the system $( A - 2I ) X = \begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}$ has
(1) no solution
(2) infinitely many solutions
(3) unique solution
(4) exactly two solutions

If the system of equations $\alpha x + y + z = 5 , x + 2 y + 3 z = 4 , x + 3 y + 5 z = \beta$ has infinitely many solutions, then the ordered pair $( \alpha , \beta )$ is equal to
(1) $( 1 , - 3 )$
(2) $( - 1,3 )$
(3) $( 1,3 )$
(4) $( - 1 , - 3 )$

The system of equations $-kx + 3y - 14z = 25$ $-15x + 4y - kz = 3$ $-4x + y + 3z = 4$ is consistent for all $k$ in the set
(1) $R$
(2) $R - \{-11, 13\}$
(3) $R - \{-13\}$
(4) $R - \{-11, 11\}$

If the system of linear equations $2x + 3y - z = -2$ $x + y + z = 4$ $x - y + |\lambda|z = 4\lambda - 4$ where $\lambda \in \mathbb{R}$, has no solution, then
(1) $\lambda = 7$
(2) $\lambda = -7$
(3) $\lambda = 8$
(4) $\lambda^2 = 1$

If the system of linear equations $2 x + y - z = 7$ $x - 3 y + 2 z = 1$ $x + 4 y + \delta z = k$, where $\delta , k \in R$ has infinitely many solutions, then $\delta + k$ is equal to
(1) $- 3$
(2) 3
(3) 6
(4) 9

The number of real values of $\lambda$, such that the system of linear equations $2x - 3y + 5z = 9$ $x + 3y - z = -18$ $3x - y + (\lambda^2 - |\lambda|)z = 16$ has no solutions, is
(1) 0
(2) 1
(3) 2
(4) 4

If the system of equations $x + y + z = 6$ $2x + 5y + \alpha z = \beta$ $x + 2y + 3z = 14$ has infinitely many solutions, then $\alpha + \beta$ is equal to
(1) 8
(2) 36
(3) 44
(4) 48

Let $N$ denote the number that turns up when a fair die is rolled. If the probability that the system of equations $x + y + z = 1$, $2 x + N y + 2 z = 2$, $3 x + 3 y + N z = 3$ has unique solution is $\frac { k } { 6 }$, then the sum of value of $k$ and all possible values of $N$ is
(1) 18
(2) 19
(3) 20
(4) 21

If a point $P(\alpha, \beta, \gamma)$ satisfying $\begin{pmatrix} \alpha & \beta & \gamma \end{pmatrix} \begin{pmatrix} 2 & 10 & 8 \\ 9 & 3 & 8 \\ 8 & 4 & 8 \end{pmatrix} = \begin{pmatrix} 0 & 0 & 0 \end{pmatrix}$ lies on the plane $2x + 4y + 3z = 5$, then $6\alpha + 9\beta + 7\gamma$ is equal to $\_\_\_\_$.

For the system of equations $x + y + z = 6$ $x + 2y + \alpha z = 10$ $x + 3y + 5z = \beta$, which one of the following is NOT true?
(1) System has no solution for $\alpha = 3, \beta = 24$
(2) System has a unique solution for $\alpha = -3, \beta = 14$
(3) System has infinitely many solutions for $\alpha = 3, \beta = 14$
(4) System has a unique solution for $\alpha = 3, \beta = 14$

Let $S$ denote the set of all real values of $\lambda$ such that the system of equations $$\lambda x + y + z = 1$$ $$x + \lambda y + z = 1$$ $$x + y + \lambda z = 1$$ is inconsistent, then $\sum_{\lambda \in S} (\lambda^2 + \lambda)$ is equal to
(1) 2
(2) 12
(3) 4
(4) 6

Let A be a symmetric matrix such that $| A | = 2$ and $\left[ \begin{array} { l l } 2 & 1 \\ 3 & \frac { 3 } { 2 } \end{array} \right] A = \left[ \begin{array} { l l } 1 & 2 \\ \alpha & \beta \end{array} \right]$. If the sum of the diagonal elements of A is $s$, then $\frac { \beta s } { \alpha ^ { 2 } }$ is equal to $\_\_\_\_$.

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

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

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