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Use product `[[1,-1, 2],[ 0, 2,-3],[ 3,-2, 4]] [[-2, 0, 1],[ 9, 2,-3],[ 6, 1,-2]]` to solve the system of equation:`x-y+2z=1`; `2y-3z=1`; `3x-2y+4z=2` |
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Answer» `[[1,-1,2],[0,2,-3],[3,-2,4]][[-2,0,1],[9,2,-3],[6,1,-2]]= [[-2-9+12,0-2+2,1+3-4],[0+18-18,0+4-3,0-6+6],[-6-18+24,0-4+4,3+6-8]]` `=[[1,0,0],[0,1,0],[0,0,1]] = I` So, this a form `A A^-1 = I`, where `A = [[1,-1,2],[0,2,-3],[3,-2,4]], A^-1 = [[-2,0,1],[9,2,-3],[6,1,-2]]` Now, for the given system of linear equation, `AX = B` `=>A A^-1 X = BA^-1` `=>X = BA^-1` Here, `X = [[x],[y],[z]] , B = [[1],[1],[2]], A^-1 = [[-2,0,1],[9,2,-3],[6,1,-2]]` `:.[[x],[y],[z]] = [[1],[1],[2]] [[-2,0,1],[9,2,-3],[6,1,-2]]` `=>[[x],[y],[z]] = [[-2+0+2],[9+2-6],[6+1-4]] ` `=>[[x],[y],[z]] = [[0],[5],[3]] ` `:. x = 0, y = 5 and z = 3` is the solution for the given equations. |
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