<s>[INST] How to determine the rank and determinant of $A$? let $A$ be $$A_{a} = \begin{pmatrix} a & 1 & 1 & 1 \\ 1 & a & 1 & 1\\ 1 & 1 & a & 1\\ 1 & 1 & 1 & a \end{pmatrix}$$
How can I calculate the rank of $A$ by the Gauss' methode and $\det A$?
 [/INST] $\det A_{a} =\det \begin{pmatrix} a & 1 & 1 & 1 \\ 1 & a & 1 & 1\\ 1 & 1 & a & 1\\ 1 & 1 & 1 & a \end{pmatrix}$
Adding all the columns to the 1st column we have,
$\det A_{a} = \det \begin{pmatrix} a+1+1+1 & 1 & 1 & 1 \\ a+1+1+1 & a & 1 & 1\\ a+1+1+1 & 1 & a & 1\\ a+1+1+1 & 1 & 1 & a \end{pmatrix}$
Taking $a+1+1+1$ common from row 1 we have,
$\det A_{a} = (a+1+1+1)\det \begin{pmatrix}1 & 1 & 1 & 1 \\ 1 & a & 1 & 1\\ 1 & 1 & a & 1\\ 1 & 1 & 1 & a \end{pmatrix}$
Now subtract the 1st row from the 2nd row 1st from the third and so on to get
$\det A_{a} = (a+1+1+1)\det \begin{pmatrix}1 & 1 & 1 & 1 \\ 0 & a & 1 & 1\\ 0 & 1 & a & 1\\ 0 & 1 & 1 & a \end{pmatrix}$
Now it will be easy.
After finding determinant of A find out the values of A for which the determinant is $0$.For those values of A find the rank of the matrix by noting the number of independent row of $A$ otherwise the rank is $4$.
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