Determinants

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

Let the system of linear equations $-x+2y-9z=7$ $-x+3y+7z=9$ $-2x+y+5z=8$ $-3x+y+13z=\lambda\,$ has a unique solution $x=\alpha\,,y=\beta\,,z=\gamma\,$. Then the distance of the point…

If $\Delta\,_{r}=\left|\begin{matrix} r & 2r-1 & 3r-2 \\ \frac{n}{2} & n-1 & a \\ \frac{1}{2}n\left(n-1\right) & \left(n-1\right)^{2} & \frac{1}{2}\left(n-1\right)\left(3n+4\right) \end{matrix}\right|$, then the value…

Let $a-2b+c=1.$ If $f\left(x\right)=\left|\begin{matrix} x+a & x+2 & x+1 \\ x+b & x+3 & x+2 \\ x+c & x+4 & x+3 \end{matrix}\right|,$ then:

If $\Delta\,_{1}=\left|\begin{matrix} x & sin\theta\, & cos\theta\, \\ -sin\theta\, & -x & 1 \\ cos\theta\, & 1 & x \end{matrix}\right|$ and $\Delta\,_{2}=\left|\begin{matrix} x & sin2\theta\, & cos2\theta\, \\…

The values of $\alpha\,$, for which$\left|\begin{matrix} 1 & \frac{3}{2} & \alpha\,+\frac{3}{2} \\ 1 & \frac{1}{3} & \alpha\,+\frac{1}{3} \\ 2\alpha\,+3 & 3\alpha\,+1 & 0 \end{matrix}\right|=0$, lie in the interval

Consider the matrices : $A=\left[\begin{array}{ll}2 & -5 \\ 3 & m\end{array}\right], B=\left[\begin{array}{l}20 \\ m\end{array}\right]$ and $X=\left[\begin{array}{l}x \\ y\end{array}\right]$. Let the set of all $m$, for…

If the system of linear equations $x-2y+z=-4$ $2x+\alpha\,y+3z=5$ $3x-y+\beta\,z=3$ has infinitely many solutions, then$12\alpha\,+13\beta\,$isequal to

For $\alpha\,,\beta\,\in\,\mathbb{R}$, suppose the system of linear equations $x-y+z=5$ $2x+2y+\alpha\,z=8$ $3x-y+4z=\beta\,$ has infinitely many solutions. Then $\alpha\,$ and $\beta\,$ are the roots of

Let $a,b,c,d$ be in arithmetic progression with common difference $\lambda\,$. If$\left|\begin{matrix} x+a-c & x+b & x+a \\ x-1 & x+c & x+b \\ x-b+d & x+d & x+c \end{matrix}\right|=2$, then value of $\lambda\,^{2}$ is…

Let$S$be the set of all$\lambda\,\in\,R$for which the system of linear equations $2x-y+2z=2$ $x-2y+\lambda\,z=-4$ $x+\lambda\,y+z=4$ has no solution. Then the set$S$

If $A=\left[\begin{matrix} 1 & \sin\,\theta\, & 1 \\ -\sin\,\theta\, & 1 & \sin\,\theta\, \\ -1 & -\sin\,\theta\, & 1 \end{matrix}\right]$, then for all…