# Find all right inverses of matrix A

$A = \left( \begin{array}{ccc} 1 & 1 & 0 \\ 2 & 3 & 1 \end{array} \right)$

I understand that this is demands solving the system

$A_{2,3} \cdot x_{3,2} = I_{2,2}$

Which can be written as such:

$x_{11} - x_{21} = 1$

$x_{12} - x_{22} = 0$

$2 x_{11} + 3 x_{21} + x_{31} = 0$

$2 x_{12} + 3 x_{22} + x_{32} = 1$

And now I need to find the what all of the solutions are, this is where I need clarification. What is the next step in finding the solutions?

Sorry if the formatting is poor, I did my best, any help is appreciated thank you!

• You have here a system of four linear equations in six variables. Solving such things is often the first thing one learns in linear algebra ... Does it confuse you that the variables have two subscripts instead of one? Ignore that; all that matters is that you have six different variables. Commented May 7, 2015 at 21:22

We wish to find all right inverses of $$A= \begin{bmatrix} 1&1&0\\2&3&1 \end{bmatrix}$$ To do so, note that $$B= \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \\ b_{31} & b_{32} \end{bmatrix}$$ satisfies $AB=I$ if and only if \begin{array}{rcrcrcrcrcrcr} b_{11} & + & b_{21} & & &&&&&&& = & 1 \\ &&&&&&b_{12} & + & b_{22} & & & = & 0 \\ 2\,b_{11} & + & 3\,b_{21} & + & b_{31} &&&&&&& = & 0 \\ &&&&&&2\,b_{12} & + & 3\,b_{22} & + & b_{32} & = & 1 \end{array} A quick row reduction then gives $$\DeclareMathOperator{rref}{rref}\rref \begin{bmatrix} 1&1&0&0&0&0&1 \\ 0&0&0&1&1&0&0\\2&3&1&0&0&0&0\\ 0&0&0&2&3&1&1 \end{bmatrix}= \begin{bmatrix} 1&0&-1&0&0&0&3\\0&1&1&0&0&0&-2\\0&0&0&1&0&-1&-1\\0&0&0&0&1&1&1 \end{bmatrix}$$ This means $AB=I$ if and only if $$B= \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \\ b_{31} & b_{32} \end{bmatrix} = \begin{bmatrix} 3+b_{31} & -1+b_{32} \\ -2-b_{31} & 1-b_{32} \\ b_{31} & b_{32} \end{bmatrix}$$ Hence all right inverses of $A$ are of the form $$B= \begin{bmatrix} 3 & -1 \\ -1 & 1 \\ 0 & 0 \end{bmatrix} + x \begin{bmatrix} 1 & 0\\ -1 & 0 \\ 1 & 0 \end{bmatrix} + y \begin{bmatrix} 0 & 1 \\ 0 & -1 \\ 0 &1 \end{bmatrix}$$ for $x,y\in\Bbb R$.