Given, "A =\\begin{bmatrix}\n4 & 4 & -2 \\\\\n-1 & 0 & 1\\\\\n3 & 6 & -1\n\\end{bmatrix}".
"A^{2}=\\begin{bmatrix}\n4 & 4 & -2 \\\\\n-1 & 0 & 1\\\\\n3 & 6 & -1\n\\end{bmatrix} \\cdot \\begin{bmatrix}\n4 & 4 & -2 \\\\\n-1 & 0 & 1\\\\\n3 & 6 & -1\n\\end{bmatrix}\\\\~\\\\~~~~~~ = \\begin{bmatrix}\n6 & 4 & -2 \\\\\n-1 & 2 & 1\\\\\n3 & 6 & 1\n\\end{bmatrix}\\\\~\\\\ ~~~~~~= \\begin{bmatrix}\n4 & 4 & -2 \\\\\n-1 & 0 & 1\\\\\n3 & 6 & -1\n\\end{bmatrix} + \\begin{bmatrix}\n2 & 0 & 0 \\\\\n0 & 2 & 0\\\\\n0 & 0 & 2\n\\end{bmatrix}\\\\~\\\\~~~~~~ = A + 2I."
Therefore, "A^{2}=A+kI" for "k=2".
Multiplying, "A^{2}=A+2I" by "A^{-1}", we get
"A = I + 2A^{-1}\\\\~\\\\\nA^{-1} = \\frac{1}{2}(A-I)\\\\~\\\\\n~~~~~~~~=\\dfrac{1}{2}\\begin{bmatrix} 3 & 4 & 2\\\\ -1 & -1 & 1 \\\\3 & 6 & -2\\end{bmatrix}"
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