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Find the general inverse (G) of the matrix $$A=\begin{bmatrix}1 & 2 & 3 \\4 & 5 & 6\end{bmatrix}$$ Also check that $AGA=A$

I am new in G- inverse calculation. I understand that G will be a $3 \times 2$ matrix. But unable to find it. Please help. Thanks in advance.

A.D
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1 Answers1

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Let $A$ be a full rank $m\times n$ matrix. By full rank we mean $\DeclareMathOperator{rank}{rank}\rank(A)=\min\{m,n\}$.

  • If $m<n$, then $A$ has a right inverse given by $$ A^{-1}_{\text{right}}=A^\top(AA^\top)^{-1} $$

  • If $m>n$, then $A$ has a left inverse given by $$ A^{-1}_{\text{left}}=(A^\top A)^{-1} A^\top $$

Our matrix $A$ is $2\times 3$ with rank two, so $A$ has a right inverse given by $$ A_{\text{right}}^{-1}= \begin{bmatrix} -17/18 & 4/9 \\ -1/9 & 1/9 \\ 13/18 & -2/9 \end{bmatrix} $$ Putting $G=A_{\text{right}}^{-1}$ then gives $$ AGA=AA_{\text{right}}^{-1}A=AI=A $$

Brian Fitzpatrick
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    if you would have given the way of finding all the generalized inverse of it would be fine..@Brian – David Mar 31 '16 at 09:54