Finding rank of a matrix depending on parameters p

Question

Hi i having a trouble with calculate rank of matrix. Matrix is: $$ \begin{matrix} 5 &p & 5 & p \\ 1&1& 1& 1\\ p&p & 2& 2\\ \end{matrix} $$ So far i get:

1)From first column substract 2nd column: $$ \begin{matrix} 5-p &p & 5 & p \\ 0&1& 1& 1\\ 0&p & 2& 2\\ \end{matrix} $$

2)From 3rd row substract 1st row: $$ \begin{matrix} 5-p &p & 5 & p \\ 0&1& 1& 1\\ -5+p&0 & -3& 2-p\\ \end{matrix} $$

What now i have to look at my paremeter p or what?

Answer 1

Hint

1. R1 $\iff $ R2
2. In the new matrix replace R2 by R2-5R1 why?
3. In the new matrix replace R3 by R3-pR1 why?

Discuss the result.

Answer 2

Interchange first and second rows and reduce:

$$\begin{pmatrix} 1&1&1&1\\ 5&p&5&p\\ p&p&2&2\end{pmatrix}\stackrel{R_2-5R_1,\,R_3-pR_1}\longrightarrow\begin{pmatrix} 1&1&1&1\\ 0&p-5&0&p-5\\ 0&0&2-p&2-p\end{pmatrix}$$

So if $\;p=2,\,5\;$ the rank is $\;2\;$, otherwise it is $\;3\;$ .