How do these two matrices relate to each other?

Question

Given $$A = \begin{pmatrix} 0&& 1&& 0&& 0 \\ 0&& 0&& 2&& 0 \\ 0&& 0&& 0&& 3\end{pmatrix}$$ and $$B = \begin{pmatrix} 0&& 0&& 0 \\ 1&& 0&& 0 \\ 0&& 1/2&& 0\\ 0&& 0&& 1/3\end{pmatrix}$$ Matrices $A$ and $B$ are: {"1 word"}.

I know that $A \cdot B = I_3$. $B$ is also the Pseudoinverse $A^+$ of $A$, but this answer doesn't fit the question, since the property has to apply to both matrices. I don't really understand what the question is looking for. Does anyone have an idea?

Answer 1

pseudoinverse

Only showing 1. and 2. using an Octave session, 3. and 4. hold for real valued symmetric matrices.

>> A = [0,1,0,0;0,0,2,0;0,0,0,3]
A =

   0   1   0   0
   0   0   2   0
   0   0   0   3

>> B = [0,0,0;1,0,0;0,1/2,0;0,0,1/3]
B =

   0.00000   0.00000   0.00000
   1.00000   0.00000   0.00000
   0.00000   0.50000   0.00000
   0.00000   0.00000   0.33333

>> A*B*A - A
ans =

   0   0   0   0
   0   0   0   0
   0   0   0   0

>> B*A*B - B
ans =

   0   0   0
   0   0   0
   0   0   0
   0   0   0

>> A*B
ans =

   1   0   0
   0   1   0
   0   0   1

>> B*A
ans =

   0   0   0   0
   0   1   0   0
   0   0   1   0
   0   0   0   1

Answer 2

Note that the SVD of matrix $\rm A$ is

$$\rm A = \mathrm I_3 \underbrace{\begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & 2 & 0 & 0\\ 0 & 0 & 3 & 0 \end{bmatrix}}_{=: \Sigma} \, \underbrace{\begin{bmatrix} 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ 1 & 0 & 0 & 0\end{bmatrix}}_{= \rm V^\top} = \mathrm I_3 \Sigma \mathrm V^\top$$

where $\rm V$ is a permutation matrix. Hence, the pseudoinverse of $\rm A$ is

$$\mathrm A^+ = \mathrm V \Sigma^+ \mathrm I_3 = \mathrm B$$