text: Depending on the book you use, the characteristic equation of a matrix A is the determinenat of A-xI or of xI-A. Sometimes the greek letter lambda is used in place of x, but I won't do that here. In your case, you are looking for the determinant of\n2-x -1 -1\n-1 2-x -1\n-1 -1 2-x\n\nIf you expand by the first row, you get\n(2-x)[(2-x)^2 -1]\n-(-1)[(-1)(2-x) -1]\n+(-1)[1-(-1)(2-x)]\n\nwhich is \n(2-x)^3-(2-x)\n-3+x -3+x\n=8-12x+6x^2-x^3\n-2+x-6+2x\n=-9x+6x^2-x^3\n\nNow, to find the eigenvalues, you set the characteristic equation to 0 to get\n0=-x(9-6x+x^2)\n=-x(x-3)^2,\n\nso x=0,3 are eigenvalues with 3 repeated.\n\nI hope this helps!