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.