Applications of Matrices: Consistency of System of Linear Equations by Rank Method

Solve the given system of homogeneous equations $2x+3y-z=0, x-y-2z=0, 3x+y+3z=0$

The augmented matrix of a system of linear equations is $\left[\begin{array}{llll}1& 2& 7& 3\\ 0& 1& 4& 6\\ 0& 0& \lambda -7& \mu +5\end{array}\right]$. The system has infinitely many solutions if

If $\rho \left(A\right)=\rho \left(\right[A\mid B\left]\right)$, then the system $Ax=B$ of linear equations is

By using Gaussian elimination method, balance the chemical reaction equation ${\mathrm{C}}_2{\mathrm{H}}_6+{\mathrm{O}}_2⟶{\mathrm{H}}_2\mathrm{O}+{\mathrm{CO}}_2$

Determine the value of $\lambda$ for which the following system of equation has a unique solution $x+y+3z=0, 4x+3y+\lambda z=0, 2x+y+2z=0$

Solve the given system of homogeneous equations $3x+2y+7z=0, 4x-3y-2z=0, 5x+9y+23z=0$

Investigate the values of $\lambda$ and $\mu$ the system of linear equations $2x+3y+5z=9, 7x+3y-5z=8, 2x+3y+\lambda z=\mu$ have an infinite number of solutions.

Investigate the values of $\lambda$ and $\mu$ the system of linear equations $2x+3y+5z=9, 7x+3y-5z=8,2x+3y+\lambda z=\mu$ have a unique solution.

Investigate the values of $\lambda$ and $\mu$ the system of linear equations $2x+3y+5z=9, 7x+3y-5z=8,2x+3y+\lambda z=\mu$ have no solution.

Find the value of $k$ for which the equation $kx-2y+z=1, x-2ky+z=-2,$$x-2y+kz=1$ have infinitely many solutions.

Find the value of $k$ for which the equation $kx-2y+z=1, x-2ky+z=-2, x-2y+kz=1$ have unique solution.

Find the value of $k$ for which the equation $kx-2y+z=1, x-2ky+z=-2$$, x-2y+kz=1$ have no solution.

Test for consistency and if possible, solve the system of equations by rank method.

$2x-y+z=2, 6x-3y+3z=6, 4x-2y+2z=4$

Test for consistency and if possible, solve the system of equations by rank method.

$2x+2y+z=5, x-y+z=1, 3x+y+2z=4$.

Test for consistency and if possible, solve the system of equations by rank method.

$3x+y+z=2, x-3y+2z=1, 7x-y+4z=5$.

Test for consistency and if possible, solve the system of equations by rank method.

$x-y+2z=2, 2x+y+4z=7, 4x-y+z=4$