[tex]A=\begin{bmatrix}1 & -2 & 1 \\2 & -1 & -1 \\ 3& -2 & -1 \\\end{bmatrix}[/tex] is the matrix such that the vector 1,1,1 is a solution to Ax=b
Let the matrix A be
[tex]\begin{bmatrix}a & b & c \\d & e & f \\ g& h & i \\\end{bmatrix}[/tex]
Then as 1,1,1 is a solution to Ax=b.
So,
[tex]\begin{bmatrix}a & b & c \\d & e & f \\ g& h & i \\\end{bmatrix}\begin{bmatrix}1 \\1 \\ 1 \\\end{bmatrix}=\begin{bmatrix}0\\0 \\ 0 \\\end{bmatrix}\\a+b+c=0\\d+e+f=0\\f+g+h=0[/tex]
Pick any values which satisfies these equations
Take
[tex]a=1,b=-2, c=1\\d=2,e=-1,f=-1\\g=3,h=-2,i=-1[/tex]
So the matrix A is
[tex]\begin{bmatrix}1 & -2 & 1 \\2 & -1 & -1 \\ 3& -2 & -1 \\\end{bmatrix}[/tex]
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