Solve System of 5 Equations with Jordan's Matrix Properties

[DanielSauza]

Hello, I've come across the following system during my finite element theory class. I'm not quite sure about how to figure out the values of u3, u4, u5, R1 and R2. I've heard something about using Jordan's matrix properties but I'm not familiar with those. How would you go about solving this system?

Sorry for my english, not my first language.

 

[Simon Bridge]

How would you normally go about solving such systems?

 

[HallsofIvy]

It is very hard to read what you have there! It appears to be
\begin{bmatrix}R_1 \\ R_2 \\ 10 \\ 0 \\ 10 \end{bmatrix}= \begin{bmatrix}8 & 0 & -5 & 0 & 0 \\ 0 & 10 & 0 & 0 & -10 \\ -5 & 0 & 18 & 7 & -20 \\ 0 & 0 & -8 & 23 & -10 \\ 0 & -10 & -20 & -10 & 40 \end{bmatrix} \begin{bmatrix}0 \\ 0 \\ u_3 \\ u_4 \\ u_5\end{bmatrix}

Is that correct? And is the right side a matrix multiplication? If so then the 5 equations are
-5u_3= R_1
-10u_5= R_2
18u_3+ 7u_4- 20u_5= 10
-8u_3+ 23u_4- 10u_5= 0 and
-20u_3- 10 u_4+ 40u_5= 0.

The first thing I notice is that the last three equation involve u_3, u_4, and u_5 without any R_1 or R_3 so can be solved as "three equations in three unknowns". Then R_1 and R_2 can be calculated from the first two equations.