Consider the system of equations : $x + ay = 0$, $y + az = 0$ and $z + ax = 0$. Then the set of all real values of $'a'$ for which the system has a unique solution is

The sum of three numbers is $6 .$ If we multiply third number by $3$ and add second number to it, we get $11$. By adding first and third numbers, we get double of the second number. Represent it algebraically and find the numbers using matrix method.

Solve system of linear equations, using matrix method.

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

Let $A$ be a $3 \times 3$ real matrix such that $A \left(\begin{array}{l}1 \\ 1 \\ 0\end{array}\right)=\left(\begin{array}{l}1 \\ 1 \\ 0\end{array}\right) ; A \left(\begin{array}{l}1 \\ 0 \\ 1\end{array}\right)=\left(\begin{array}{c}-1 \\ 0 \\ 1\end{array}\right)$ and $A \left(\begin{array}{l}0 \\ 0 \\ 1\end{array}\right)=\left(\begin{array}{l}1 \\ 1 \\ 2\end{array}\right)$. If $X =\left( x _{1}, x _{2}, x _{3}\right)^{ T }$ and $I$ is an identity matrix of order $3$ , then the system $( A -2 I ) X =\left(\begin{array}{l}4 \\ 1 \\ 1\end{array}\right)$ has

Solve the system of the following equations  $\frac{2}{x}+\frac{3}{y}+\frac{10}{z}=4$ ; $\frac{4}{x}-\frac{6}{y}+\frac{5}{z}=1$  ; $\frac{6}{x}+\frac{9}{y}-\frac{20}{z}=2$