Applications of Determinants and Matrices

If the system $\left[\begin{array}{ll}2& 8\\ 3& 7\end{array}\right]\left[\begin{array}{l}a\\ b\end{array}\right]=k\left[\begin{array}{l}a\\ b\end{array}\right]$ has a non trivial solution, then the positive value of $K$ and $a$ solution of the system for that value of $K$ are

If the system of equations $x+ay=0, az+y=0$ and $ax+z=0$ has infinitely many solutions for some real value of $a$ and $a=(\lambda -4),$ then the value of $\lambda$ is

 Let the two matrices $A$ and $B$ be given by $A=\left[\begin{array}{ccc}1& -1& 0\\ 2& 3& 4\\ 0& 1& 2\end{array}\right]$ and $B=\left[\begin{array}{ccc}2& 2& -4\\ -4& 2& -4\\ 2& -1& 5\end{array}\right]$

Verify that $AB=BA=6I$, where $I$ is the unit matrix of order 3 and hence solve the system of equation $x-y=3$, $2x+3y+4z=17$ and $y+2z=7$.

If the solution of the system of equations is $x=a, y=b, z=c$, then $a+b+c=$

Solve system of linear equations, using matrix method.

$x-y+2z=7$

$3x+4y-5z=-5$

$2x-y+3z=12$

Solve system of linear equations, using matrix method.

$2x+3y+3z=5$

$x-2y+z=-4$

$3x-y-2z=3$. Then find $x+y+z$.

Solve system of linear equations, using matrix method.

$5x+2y=3$

$3x+2y=5$

If the solution is $x=a, y=b$, then $a+b=$

Examine the consistency of the system of equations.

$5x+2y=4$

$7x+3y=5$

If the system of equations $x + ay = 0$, $az + y= 0$ and $ax + z = 0$ has infinitely many solutions for real some value of $a$ and $a=(\lambda -4),$ then the value of $\lambda$ is

If $\theta \in \left(0,\frac{\pi }{2}\right)$ and $p=4{\mathrm{s}\mathrm{e}\mathrm{c}}^2\theta +9\mathrm{c}\mathrm{o}\mathrm{s}e{c}^2\theta$ , then the system of equations $3x-y+4z=3,x+2y-3z=-2,6x+5y+pz=-3$ have

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

The system $2x+3y+z=5, 3x+y+5z=7, x+4y-2z=3$ has

The system of equations

$\begin{array}{c}6x+5y+\lambda z=0\\ 3x-y+4z=0\\ x+2y-3z=0\end{array} has$

The system of equations $\mathrm{\alpha x}+y+z=\alpha -1,x+\alpha y+z=\alpha -1,x+y+\alpha z=\alpha -1$ has no solution, if $\alpha$ is

If the system of equation

$2\mathrm{x}\mathrm{ }+\mathrm{ }5\mathrm{y}\mathrm{ }+\mathrm{ }8\mathrm{z}=\mathrm{ }0$

$\mathrm{x}\mathrm{ }+\mathrm{ }4\mathrm{y}\mathrm{ }+\mathrm{ }7\mathrm{z}=\mathrm{ }0$

$6\mathrm{x}\mathrm{ }+\mathrm{ }4\mathrm{y}-\mathrm{\lambda }\mathrm{z}=\mathrm{ }0$

has a non trivial solution then $\mathrm{\lambda }$ is equal to -

Consider the system of linear equation in $x,\mathrm{ }\mathrm{y},\mathrm{ }\mathrm{z}$

$\left(\sin 3\theta \right)x-y+z=0$

$\left(\cos 2\theta \right)\mathrm{ }x+4y+3z=0$

$2x+7y+7z=0$

If the system has non-trivial solution then, $\theta \in \left[0,\mathrm{\pi }\right]$ are 

Consider the system of linear equation in $\mathrm{x},\mathrm{ }\mathrm{y}$ and $\mathrm{z}$

$\left(\sin 3\mathrm{\theta }\right)\mathrm{x}-\mathrm{ }\mathrm{y}\mathrm{ }+\mathrm{ }\mathrm{z}=0$

$\left(\cos 2\mathrm{\theta }\right)\mathrm{ }\mathrm{x}\mathrm{ }+\mathrm{ }4\mathrm{y}\mathrm{ }+\mathrm{ }3\mathrm{z}=0$

$2\mathrm{x}\mathrm{ }+\mathrm{ }7\mathrm{y}\mathrm{ }+\mathrm{ }7\mathrm{z}=0$

If the system has non-trivial solution, then $\mathrm{\theta }\in \left[0,\mathrm{ }\mathrm{\pi }\right]$ are -

The system of equations $\mathrm{x}\mathrm{ }+\mathrm{ }\mathrm{k}\mathrm{y}\mathrm{ }+\mathrm{ }3\mathrm{z}=0,\mathrm{ }3\mathrm{x}\mathrm{ }+\mathrm{ }\mathrm{k}\mathrm{y}\mathrm{ }-2\mathrm{z}=0,\mathrm{ }2\mathrm{x}\mathrm{ }+\mathrm{ }3\mathrm{y}\mathrm{ }-\mathrm{ }4\mathrm{z}=0$ possess a non-trivial solution over the set of rationals, then $2\mathrm{k}$ , is an integral element of the interval :

The system of equations

$-\mathrm{ }2\mathrm{x}+\mathrm{y}+\mathrm{z}=\mathrm{a}$

$\mathrm{x}-\mathrm{ }2\mathrm{y}+\mathrm{z}=\mathrm{b}$

$\mathrm{x}+\mathrm{y}-2\mathrm{z}=\mathrm{c}$

has:

If the equations $2x+3y=1, x+2y=2$ and $x-y+2\lambda =0$ have a unique solution. The $2\lambda –6$ is equal to :

The system of equations $x+2y+3z=1,$ $2x+y+3z=2$ and $5x+5y+9z=5$ has