<s>[INST] Compute Moore-Penrose Pseudoinverse of a 3x3 matrix 
Given a 3x3 matrix A= \begin{pmatrix}
1 & 1 & 1\\ 
1 & 1 &1 \\ 
1 & 1 & 1
\end{pmatrix} Find Psuedoinverse of the above matrix

For the above matrix I am getting it as singular.
Also the rank of the above matrix is 1 so tried to calculate left and right inverse of A but am getting stuck at calculating them as they are coming out to be singular.
Does the psuedoinverse of the matrix exists or the procedure followed to find it is incorrect.
 [/INST] Note that, if $A$ is the given matrix, then $A^2 = 3A$, and $A^3 = 9A$. Also, $A$ is symmetric. From this information, I claim that $B = \frac{1}{9}A$ is the pseudoinverse of $A$. Note that
\begin{align*}
ABA &= \frac{1}{9}A^3 = A \\
BAB &= \frac{1}{9^2}A^3 = \frac{1}{9}A = B \\
AB &= \frac{1}{3}A \\
BA &= \frac{1}{3}A,
\end{align*}
where $\frac{1}{3} A$ is symmetric. Hence, $B$ is the pseudoinverse of $A$.
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