Answer to Question #278751 in Linear Algebra for Petra

Question #278751

Compute the determinant, and use Gauss-Jordan elimination to find the inverse of the following matrix (if it exists).

218        0             0

0             218        2

0             1             1



1
Expert's answer
2022-01-03T19:19:55-0500

"\\begin{vmatrix}\n 218 & 0&0 \\\\\n 0 & 218&2\\\\\n0&1&1\n\\end{vmatrix}=218\\cdot 218\\cdot1+0\\cdot0\\cdot0+\\\\+0\\cdot2\\cdot0\n-0\\cdot218\\cdot0-0\\cdot0\\cdot1-\\\\\n-1\\cdot2\\cdot218=47524-436=47088\\neq0"

"A^{-1}\\\\\n\\begin{pmatrix}\n 218 &0&0 &|&1&0&0\\\\\n 0 & 218&2&|&0&1&0\\\\\n0&1&1&|&0&0&1\n\\end{pmatrix}\\sim\\\\\n1r\\cdot \\frac{1}{218}\\\\\n2r\\leftrightarrow3r"

"\\sim\\begin{pmatrix}\n 1 &0&0 &|&\\frac{1}{218}&0&0\\\\\n 0&1&1&|&0&0&1\\\\\n0 & 218&2&|&0&1&0\n\\end{pmatrix}\\sim\\\\\n3r+2r\\cdot(-218)"

"\\sim\\begin{pmatrix}\n 1 &0&0 &|&\\frac{1}{218}&0&0\\\\\n 0&1&1&|&0&0&1\\\\\n0 & 0&-216&|&0&1&-218\n\\end{pmatrix}\\sim\\\\\n3r\\cdot\\frac{-1}{216}"

"\\sim\\begin{pmatrix}\n 1 &0&0 &|&\\frac{1}{218}&0&0\\\\\n 0&1&1&|&0&0&1\\\\\n0 & 0&1&|&0&\\frac{-1}{216}&\\frac{109}{108}\n\\end{pmatrix}\\sim\\\\\n2r+3r\\cdot(-1)"


"\\sim\\begin{pmatrix}\n 1 &0&0 &|&\\frac{1}{218}&0&0\\\\\n 0&1&0&|&0&\\frac{1}{216}&\\frac{-1}{108}\\\\\n0 & 0&1&|&0&\\frac{-1}{216}&\\frac{109}{108}\n\\end{pmatrix}"


"A^{-1}=\\begin{pmatrix}\n \\frac{1}{218}&0&0\\\\\\\\\n 0&\\frac{1}{216}&\\frac{-1}{108}\\\\\\\\\n0&\\frac{-1}{216}&\\frac{109}{108}\n\\end{pmatrix}"


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