I'm given the matrix A where it's first row is $(2, -1, 3)$ and second row is $(1, 2, 1)$ and I'm told to find all the right inverses of it. First I tried doing A times a 3x2 vector B (just a vector with inputs $a, b, c, d, e, f$) and multiplying that out to result in the identity matrix. This resulted in my matrix -> first row+ $(2a-c+3e, 2b-d+3f)$ and second row= (a+2c+e, b+2d+f) where this is all equal to the identity matrix. I tried doing systems of equations but that doesn't really tell me anything except for what $a,b,...e,f$ are in terms of each other, which is a bit useless. What should I be doing?
2026-03-30 17:37:05.1774892225
Find all right inverses of matrix A.
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You want to solve the system $$ \begin{bmatrix} 2 & -1 & 3 \\ 1 & 2 & 1 \end{bmatrix} \begin{bmatrix} x_{11} & x_{12} \\ x_{21} & x_{22} \\ x_{31} & x_{32} \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} $$ This can be written as two linear systems: $$ \begin{cases} 2x_{11}-x_{21}+3x_{31}=1\\ x_{11}+2x_{21}+x_{31}=0 \end{cases} \qquad \begin{cases} 2x_{12}-x_{22}+3x_{32}=0\\ x_{12}+2x_{22}+x_{32}=1 \end{cases} $$ Can you find all solutions of those systems?