If the system of linear equations $$2x+3y-z=-2$$ $$x+y+z=4$$ $$x-y+|\lambda |z=4\lambda -4$$ where, $$\lambda$$ $$\in$$ R, has no solution, then

Let A be a matrix of order 3 $$\times$$ 3 and det (A) = 2. Then det (det (A) adj (5 adj (A3))) is equal to _____________.

Let f(x) = \left| {\matrix{ a & { - 1} & 0 \cr {ax} & a & { - 1} \cr {a{x^2}} & {ax} & a \cr } } \right|,\,a \in R. Then the sum of the squares of all the values of a, for which $$2f^{\prime }(10)-f^{\prime }(5)+100=0$$, is

Let A and B be two 3 $$\times$$ 3 matrices such that $$AB=I$$ and |A| = {1 \over 8}. Then $$|adj(Badj(2A))|$$ is equal to

Let the system of linear equations $$x+2y+z=2$$, $$\alpha x+3y-z=\alpha$$, $$-\alpha x+y+2z=-\alpha$$ be inconsistent. Then $$\alpha$$ is equal to :

Let A be a 3 $$\times$$ 3 invertible matrix. If |adj (24A)| = |adj (3 adj (2A))|, then |A|2 is equal to :

The ordered pair (a, b), for which the system of linear equations 3x $$-$$ 2y + z = b 5x $$-$$ 8y + 9z = 3 2x + y + az = $$-$$1 has no solution, is :

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

The system of equations $$-kx+3y-14z=25$$ $$-15x+4y-kz=3$$ $$-4x+y+3z=4$$ is consistent for all k in the set

Let A be a 3 $$\times$$ 3 real matrix such that $$A\left(\text{matrix}1\text{cr}1\text{cr}0\text{cr}\right)=\left(\text{matrix}1\text{cr}1\text{cr}0\text{cr}\right);A\left(\text{matrix}1\text{cr}0\text{cr}1\text{cr}\right)=\left(\text{matrix}-1\text{cr}0\text{cr}1\text{cr}\right)$$ and $$A\left(\text{matrix}0\text{cr}0\text{cr}1\text{cr}\right)=\left(\text{matrix}1\text{cr}1\text{cr}2\text{cr}\right)$$. If $$X=(x_1,x_2,x_3{)}^T$$ and I is an identity matrix of order 3, then the system $$(A-2I)X=\left(\text{matrix}4\text{cr}1\text{cr}1\text{cr}\right)$$ has :

Let A = \left[ {\matrix{ 0 & { - 2} \cr 2 & 0 \cr } } \right]. If M and N are two matrices given by $$M=\sum _{k=1}^{10}{A}^{2k}$$ and $$N=\sum _{k=1}^{10}{A}^{2k-1}$$ then MN2 is :

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

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

Let S = {$$\sqrt{n}$$ : 1 $$\le$$ n $$\le$$ 50 and n is odd}. Let a $$\in$$ S and A = \left[ {\matrix{ 1 & 0 & a \cr { - 1} & 1 & 0 \cr { - a} & 0 & 1 \cr } } \right]. If $$\sum _{a\in S}\det (adjA)=100\lambda$$, then $$\lambda$$ is equal to :

Let A = \left[ {\matrix{ 1 & { - 2} & \alpha \cr \alpha & 2 & { - 1} \cr } } \right] and B = \left[ {\matrix{ 2 & \alpha \cr { - 1} & 2 \cr 4 & { - 5} \cr } } \right],\,\alpha \in C. Then the absolute value of the sum of all values of $$\alpha$$ for which det(AB) = 0 is :

Let A and B be two square matrices of order 2. If $$det(A)=2$$, $$det(B)=3$$ and $$\det \left((\det 5(detA)B)A^2\right)=2^a{3}^b{5}^c$$ for some a, b, c, $$\in$$ N, then a + b + c is equal to :

The number of $$\theta \in (0,4\pi )$$ for which the system of linear equations $$\begin{array}{rl}& 3(\sin 3\theta )x-y+z=2\\ & \\ & 3(\cos 2\theta )x+4y+3z=3\\ & \\ & 6x+7y+7z=9\end{array}$$ has no solution, is :

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

If the system of linear equations. $$8x+y+4z=-2$$ $$x+y+z=0$$ $$\lambda x-3y=\mu$$ has infinitely many solutions, then the distance of the point \left( {\lambda ,\mu , - {1 \over 2}} \right) from the plane $$8x+y+4z+2=0$$ is :

Let A be a 2 $$\times$$ 2 matrix with det (A) = $$-$$ 1 and det ((A + I) (Adj (A) + I)) = 4. Then the sum of the diagonal elements of A can be :