Question

Use cofactors to compute the inverse of the following matrix: A:1 -2 1. し0-12

Answer 1

Given A = [2 1 0 1 -2 -1 0 1 2] A^-1 = 1/ Vertical-bar A Vertical-bar (cofactor of A)^T Vertical-bar A Vertical-bar = 2(-2 times 2 - (-1 times 1) - 1(1 times 2 - 1 times 0) + 0(1 times -1 - (-2 times 0)) = 2(-4 + 1) - 1(2) = 2(-3) - 1(2) = -6 - 2 = -8 A 3 times 3 matrix can be generally represented as [a_11 a_21 a_31 a_12 a_22 a_32 a_13 a_23 a_33] where suffixes of a are represented as a_ij cofactor of element can be written as co f(a_11) = (-1)^1 + 1 Vertical-bar a_22 a_32 a_23 a_33 Vertical-bar the elements inside determinant are those excluding row and column of a_11 \r\n co f(a_11) = 1 Vertical-bar -2 -1 1 2 Vertical-bar = -4 + 1 = -3 co f(a_12) = (-1)^1 + 2 Vertical-bar 1 1 2 0 Vertical-bar = (-1)(-2) = 2 co f(a_13) = (-1)^1 + 3 Vertical-bar 1 0 -2 -1 Vertical-bar = 1. (-1) = -1 co f(a_21) = (-1)^2 + 1 Vertical-bar 1 -1 0 2 Vertical-bar = (-1)(2) = -2