rank of the matrix dependent on parameter

Question

I want to find rank of the matrix: $$\begin{pmatrix} 0 & p & 6 & -p & p+2\\ 3-p & p-2 & p-1 & -1 & 1\\ -4 & 2 & 0 & 2 & p-3 \end{pmatrix}$$ I changed 1st with 3rd row, and then I tried doing some changes between rows but nothing got me anywhere... Could anyone help with 2/3 first steps? I have test from this today :/

Answer 1

The rank of a matrix is always less than or equal the minimum of the number of rows and of columns. In this case, $\;3\;$ , so $\;rk(A)\le3\;$ . Take now the first three columns and form a square matrix with them and evaluate its determinant:

$$\begin{vmatrix}0&p&6\\ 3-p&p-2&p-1\\ -4&2&0\end{vmatrix}=-4p(p-1)+12(3-p)+24(p-2)=-4p^2+16p-12\stackrel?=0\iff$$

$$p^2-4p+3=0\iff(p-3)(p-1)=0$$

and thus for $\;p\neq1,3\;$ the matrix has rank three. Now you can check what happens with the two particular cases $\;p=1,3\;$ by substituting and, say. reducing (by rows or by columns: it will give the same result for the rank, of course).