# Solution to matrix equation

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1. Sep 3, 2015

### kostoglotov

Hi, rapid fire posting in this subforum I know, sorry if that's annoying. Let me know if I should space my posts out a bit more.

Here's an image of the solution to a worked example (from Intro to Linear Algebra 4th by Strang)

I cannot figure out what the underlined (underlining added by me) sentences mean:

"The first and last rows give x2 and x3. Then the middle rows give x1 and x4."

How do the middle rows only give x1 and x4? Is this the result of treating this as a system of simultaneous equations rather than linear combinations? That's the only explanation I can find at the moment. Mostly I don't know what the author's intended meaning is.

I could find all the x's by solving simultaneous equations, but would that be how the author has done it, or has Strang found the x's a different way?

2. Sep 3, 2015

### BvU

What the author describes IS the solving of the 'simultaneous' equations :
$x_2 = b_1 \\ x_3-x_1 = b_2\\ x_4 - x_2 = b_3\\ -x_3 = b_4$​

we humans don't do 'simultaneously: we see that this means that

first and last rows give x2 and x3 and THEN the middle rows give x1 and x4.​

Thereby Cx = b has been solved, and that can only be the case if C is invertible (x = C-1 b).

3. Sep 3, 2015

### Orodruin

Staff Emeritus
Solving the system of simultaneous equations just means that the same solution should be valid for all equations.