System of linear equations

[diogomgf]

I've checked and re-checked this exercise and I can't figure out where things have gone wrong:

For [MATH]a \in R[/MATH] discuss the following system of linear equations:
[MATH]\left\{\begin{matrix} (8-a)x + 2y + 3z + aw = 0\\ x + (9-a)y + 4z + aw = 0\\ x + 2y + (10-a)z + aw = 0\\ x + 2y + 3z + aw = 0 \end{matrix}\right.[/MATH]
And this is how I solved it with gauss-jordan elimination:

 

[Steven G]

I would be willing to follow your work but not if you are not going to tell me the rules that you used to go from one matrix to another.

According to you, you made an error somewhere. Do you realize how painful it can be to figure what your rule(s) from one matrix to another if there is an error made. These are your rules and I am not a mind reader! Please state the rules and post back. Also why do you think you made a mistake?

 

[diogomgf]

I would be willing to follow your work but not if you are not going to tell me the rules that you used to go from one matrix to another.

According to you, you made an error somewhere. Do you realize how painful it can be to figure what your rule(s) from one matrix to another if there is an error made. These are your rules and I am not a mind reader! Please state the rules and post back. Also why do you think you made a mistake?

Either me or the book's solutions are wrong :\

Sorry for not posting the transformations earlier:

[MATH]\underset{T_{1}}{\rightarrow} (L_{2}, L_{3}, L_{4}) - L_{1}[/MATH][MATH]\underset{T_{2}}{\rightarrow} (L_{2}, L_{3}, L_{4}) / (a-7)[/MATH][MATH]\underset{T_{3}}{\rightarrow} L_{4} \Leftrightarrow L_{1} [/MATH][MATH]\underset{T_{4}}{\rightarrow} (L_{2}, L_{3}) - L_{1} , L_{4}) + (a-8)L_{1}[/MATH][MATH]\underset{T_{5}}{\rightarrow} L_{2} + (\frac{1}{a-7})L_{3}, L_{4} + 3L_{3}[/MATH][MATH]\underset{T_{6}}{\rightarrow} -(L_{2},L_{3}), \frac{L_{4}}{a}[/MATH]

 

[Cubist]

Those steps look reasonable to me. What does the final result imply?

Also are there any values of \(\displaystyle a\) for which steps T2 and T6 don't hold? What happens if \(\displaystyle a\) has these particular values? Does this explain the difference between your work and the answer at the back of the book?

 

[Romsek]

[MATH]\begin{pmatrix}x\\y\\z\\w\end{pmatrix}=\begin{pmatrix}0\\0\\0\\0\end{pmatrix}[/MATH]
is the correct answer

 

[diogomgf]

Those steps look reasonable to me. What does the final result imply?

Also are there any values of \(\displaystyle a\) for which steps T2 and T6 don't hold? What happens if \(\displaystyle a\) has these particular values? Does this explain the difference between your work and the answer at the back of the book?

Yup if a =7 or a = 0 then those steps don't hold. That's what I was not understanding in the books solutions.

[MATH]\begin{pmatrix}x\\y\\z\\w\end{pmatrix}=\begin{pmatrix}0\\0\\0\\0\end{pmatrix}[/MATH]
is the correct answer

It's partially correct, as pointed by Cubist

 

[Romsek]

Yup if a =7 or a = 0 then those steps don't hold. That's what I was not understanding in the books solutions.

It's partially correct, as pointed by Cubist

I plugged the system into Mathematica and got the zero vector as an answer irregardless of the value of a.

I'll take a second look.

Cubist is correct. Very odd that Mathematica just spit out the zero vector as the solution. I'll have to investigate that further.