Systems of linear equations and inequalities

Find the values of a real parameter $t$ which ensure that the following system has a unique solution $[a,b]$ such that both $a$ and $b$ are positive real numbers. $$\begin{array}{rlrlrlrlrlr}a& -& tb& =-& 2& & & & & & \\ a& +2& tb& =& 0& & & & & & \end{array}$$

The coefficient matrix of a $3\times 3$ linear system is $A$ and the augmented matrix $A^{\prime }$. Find $\mathrm{rank}(A)$ and $\mathrm{rank}(A^{\prime })$. $$A=\left(\begin{array}{ccc}-1& 3& 2\\ 0& 4& -5\\ 0& 0& 2\end{array}\right)A^{\prime }=\left(\begin{array}{cccc}-1& 3& 2& 5\\ 0& 4& -5& 10\\ 0& 0& 2& 0\end{array}\right)$$

Consider a linear system of four equations with four unknowns. The rank of the coefficient matrix $A$ is $\mathrm{rank}(A)=3$. The rank of the augmented matrix $A^{\prime }$ is $\mathrm{rank}(A^{\prime })=4$. Identify a true statement on this system.

The augmented matrix of a system of three equations with three unknowns is row equivalent with the following matrix $A^{\prime }$. Find the solution of the system. $$A^{\prime }=\left(\begin{array}{cccc}-1& 0& 1& -1\\ 0& 7& 2& -1\\ 0& 0& 30& 6\end{array}\right)$$