find two different generalized inverse of the given matrix

If $AGA=A$, then $A(G+uv^T)A=A$ if $u\in\ker A$ or $v\in\ker A^T$. Note that when $A$ is not a nonsingular matrix (this includes the case where $A$ is not square), at least one of $A$ or $A^T$ has a nonzero nullspace. Therefore, if you can find one generalised inverse of $A$, you can find infinitely many others if the field is infinite.

By the way, the two matrices that you claim to be generalised inverses of your example $A$ do not seem to be correct.

Tags:

Linear Algebra

Matrices

Pseudoinverse

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