linear algebra - To find eigenvalues of matrix with all same element

How many distinct eigenvalues are there in the matrix.

$$\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ \end{bmatrix}$$

I was wondering that is there any specific eigenvalues for matrices like this??

I hope I wouldn't have to find the determinant of this 4×4 matrix.

Answer

Note that $$\left(\begin{matrix}1&1&1&1\\ 1&1&1&1\\ 1&1&1&1\\ 1&1&1&1\end{matrix}\right)\left(\begin{matrix} a\\ b\\ c\\ d\end{matrix}\right)=\left(\begin{matrix} a+b+c+d\\ a+b+c+d\\ a+b+c+d\\ a+b+c+d\end{matrix}\right)$$
If $a=b=c=d$, then $$\left(\begin{matrix}1&1&1&1\\ 1&1&1&1\\ 1&1&1&1\\ 1&1&1&1\end{matrix}\right)\left(\begin{matrix} a\\ a\\ a\\ a\end{matrix}\right)=4\left(\begin{matrix} a\\ a\\ a\\ a\end{matrix}\right)$$
Hence $4$ is an eigenvalue.

Also see that since all the columns of the matrix are same, the rank of the matrix is $1$. So $4$ is the only non zero eigenvalue. $0$ is the other eigenvalue, with eigenvector, for example $(a~-a~a~-a)^t$.