Answer to Question #274150 in Linear Algebra for Nikhil Singh

Question #274150

Let T: R^3 to R^3 be the linear transformation defined by


T(x,y,z)= (-x,x-y, 3x+2y+z)


Check whether T satifies the polynomial (x-1)(x+1)^2. Also the find of minimal polynomial of T.

1
Expert's answer
2022-01-18T11:23:40-0500

"T=x\\begin{pmatrix}\n -1 \\\\\n 1\\\\\n3\n\\end{pmatrix}+y\\begin{pmatrix}\n 0 \\\\\n -1\\\\\n2\n\\end{pmatrix}+z\\begin{pmatrix}\n 0 \\\\\n 0\\\\\n1\n\\end{pmatrix}"


"T=\\begin{pmatrix}\n -1&0 & 0 \\\\\n 1&-1 & 0\\\\\n3&2&1\n\\end{pmatrix}"




"T-1=\\begin{pmatrix}\n -2&0& 0 \\\\\n 1&-2 & 0\\\\\n3&2&0\n\\end{pmatrix}"



"(T+1)^2=\\begin{pmatrix}\n 0&0&0 \\\\\n 1&0&0\\\\\n3&2&2\n\\end{pmatrix}^2=\\begin{pmatrix}\n 0&0&0\\\\\n 0&0& 0\\\\\n8&4&4\n\\end{pmatrix}"



"(T-1)(T+1)^2=\\begin{pmatrix}\n -2&0& 0 \\\\\n 1&-2 & 0\\\\\n3&2&0\n\\end{pmatrix}\\begin{pmatrix}\n 0&0&0 \\\\\n 0&0 & 0\\\\\n8&4&4\n\\end{pmatrix}=\\begin{pmatrix}\n 0&0& 0 \\\\\n 0&0& 0\\\\\n0&0&0\n\\end{pmatrix}"




T satisfies the polynomial "(x-1)(x+1)^2"


"(T-1)(T+1)=\\begin{pmatrix}\n -2&0&0 \\\\\n 1&-2 & 0\\\\\n3&2&0\n\\end{pmatrix}\\begin{pmatrix}\n 0&0& 0\\\\\n 1&0&0 \\\\\n3&2&2\n\\end{pmatrix}=\\begin{pmatrix}\n 0&0 & 0\\\\\n -2&0&0\\\\\n2&0&0\n\\end{pmatrix}\\ne\\begin{pmatrix}\n 0&0 & 0 \\\\\n 0&0& 0\\\\\n0&0&0\n\\end{pmatrix}"




"\\therefore" Minimal polynomial is "(x-1)(x+1)^2"

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