Solving Linear Equations with augmented matrices

Question

Could someone please walk me through this question as I am struggling with it.

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Answer 1

Form the augmented coefficients matrix and begin reducing it by rows. Observe that I already interchanged rows 1 and 2:

$$\begin{pmatrix}1&p&1&|&p\\ p&1&1&|&1\\ 1&1&p&|&p^2\end{pmatrix}\longrightarrow\begin{pmatrix}1&p&1&|&p\\ 0&1-p^2&1-p&|&1-p^2\\ 0&1-p&p-1&|&p^2-p\end{pmatrix}\;(**)$$

You can now check directly for $\;p=-1\;$ , and then for $\;p\neq-1\;$ you can add the second row multiplied by $\;-\cfrac1{1+p}\;$ to the third row and get::

$$(**)\longrightarrow\begin{pmatrix}1&p&1&|&p\\ 0&1-p^2&1-p&|&1-p^2\\ 0&0&(p+2)(p-1)&|&p^2-1\end{pmatrix}$$

Well, what can you see and deduce from up there?