If the system of linear equation $3 x-2 y+z=2, 4 x-3 y+3 z=-5$ and $7 x-5 y+\lambda z=9$ has no solution, then $\lambda$ equals to

Let $A=\left[\begin{array}{lll}3 & 2 & 3 \\ 4 & 1 & 0 \\ 2 & 5 & 1\end{array}\right]$ and $49 B=\left[\begin{array}{ccc}1 & 13 & -3 \\ -4 & -3 & 12 \\ \alpha & -11 & -5\end{array}\right]$ If $B$ is the inverse of $A$, then the value of $\alpha$ is

$ \text { If } A=\left[\begin{array}{cc} \sin \theta & -\cos \theta \\ \cos \theta & \sin \theta \end{array}\right] \text {, then } A(\operatorname{adj} A)^{-1} \text { equals to } $

If $a, b, c$ are non-zero real numbers and if the system of equations $(a-1) x-y-z=0, -x+(b-1) y-z=0,-x-y+(c-1) z=0$ has a non-trivial solution, then $a b+b c+c a$ equals to

Given 2x $-$ y + 2z = 2, x $-$ 2y - z = $-$4, x + y + $\lambda$z = 4, then the value of $\lambda$ such that the given system of equation has no solution is

Let $A = \left[ {\matrix{ 1 & { - 1} & 1 \cr 2 & 1 & { - 3} \cr 1 & 1 & 1 \cr } } \right]$ and $10B = \left[ {\matrix{ 4 & 2 & 2 \cr { - 5} & 0 & \alpha \cr 1 & { - 2} & 3 \cr } } \right]$

If B is the inverse of A, then the value of $\alpha$ is

If $\left[ {\matrix{ 1 & { - \tan \theta } \cr {\tan \theta } & 1 \cr } } \right]{\left[ {\matrix{ 1 & {\tan \theta } \cr { - \tan \theta } & 1 \cr } } \right]^{ - 1}} = \left[ {\matrix{ a & { - b} \cr b & a \cr } } \right]$, then

If p $\ne$ a, q $\ne$ b, r $\ne$ c and the system of equations

px + ay + az = 0

bx + qy + bz = 0

cx + cy + rz = 0

has a non-trivial solution, then the value of $\frac{p}{p-a}+\frac{q}{q-b}+\frac{r}{r-c}$ is

Matrix $A = \left| {\matrix{ x & 3 & 2 \cr 1 & y & 4 \cr 2 & 2 & z \cr } } \right|$, if xyz = 60 and 8x + 4y + 3z = 20, then A(adj A) is equal to

An ordered pair ($\alpha$, $\beta$) for which the system of linear $(1 + \alpha )x + \beta y + z = 2$, $\alpha x + (1 + \beta )y + z = 3$, $\alpha x + \beta y + 2z = 2$ has a unique solution.

Consider matrix $A = \left[ {\matrix{ 2 & 1 \cr 1 & 2 \cr } } \right]$, if ${A^{ - 1}} = \alpha I + \beta A$, where $\alpha$, $\beta$ $ \notin $ R, then ($\alpha$ + $\beta$) is equal to (where A^$-$1 denotes the inverse of matrix A)