Step 1: Identify the matrix J, which is J = [[1, 2, 1], [0, 1, 0], [2, 1, 1]].
Step 2: Write down the formula for the determinant of a 3x3 matrix: det(J) = a(ei - fh) - b(di - fg) + c(dh - eg), where the matrix is [[a, b, c], [d, e, f], [g, h, i]].
Step 3: Assign values from matrix J to the variables: a = 1, b = 2, c = 1, d = 0, e = 1, f = 0, g = 2, h = 1, i = 1.
Step 4: Calculate the first part: ei - fh = (1*1) - (0*1) = 1 - 0 = 1.
Step 5: Calculate the second part: di - fg = (0*1) - (0*2) = 0 - 0 = 0.
Step 6: Calculate the third part: dh - eg = (0*1) - (1*2) = 0 - 2 = -2.
Step 7: Substitute these values back into the determinant formula: det(J) = 1(1) - 2(0) + 1(-2).
