(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show that the system ∑ hereunder admits one unique solution

[tex]

∑ =

\left[\begin{array}{cc}

1 & a_{1} & a_{1}{}^{2} & a_{1}{}^{3} & | & b_{1}\\

1 & a_{2} & a_{2}{}^{2} & a_{2}{}^{3} & | & b_{2}\\

1 & a_{3} & a_{3}{}^{2} & a_{3}{}^{3} & | & b_{3}\\

1 & a_{4} & a_{4}{}^{2} & a_{4}{}^{3} & | & b_{4}

\end{array}\right]

[/tex]

3. The attempt at a solution

I know I have to perform Gauss-Jordan elimination fully by pivoting on the diagonal and zeroing out both on top and bottom of the diagonal.

After just two steps I'm stuck. I came up with this:

[tex]

∑ ∼

\left[\begin{array}{cc}

1 & a_{1} & a_{1}{}^{2} & a_{1}{}^{3} & | & b_{1}\\

0 & a_{2} - a_{1} & a_{2}{}^{2} - a_{1}{}^{2} & a_{2}{}^{3} - a_{1}{}^{3} & | & b_{2} - b_{1}\\

0 & 0 & - & - & | & -\\

0 & 0 & - & - & | & -

\end{array}\right]

[/tex]

I don't know how to continue to fill in the _ spaces as I don't know what operation zeroed out the values under the pivot.

I don't like dividing the pivots through just so they're of value 1 as to avoid working with fractions.

I know I'm looking for a compatible system (i.e. (n - r) = 0, n being #rows and r being the rank) that also gives 0 secondary unknowns (i.e. (p - r) = 0, p being #columns not including the RHS and r = rank).

Thank you very much!

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: LINEAR ALGEBRA: How to prove system has one unique solution

**Physics Forums | Science Articles, Homework Help, Discussion**