If J = [[1, 2, 1], [0, 1, 3], [2, 1, 0]], calculate det(J). (2023)

If J = [[1, 2, 1], [0, 1, 3], [2, 1, 0]], calculate det(J). (2023)

Step 1: Identify the matrix J. J = [[1, 2, 1], [0, 1, 3], [2, 1, 0]].
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 = 3, g = 2, h = 1, i = 0.
Step 4: Calculate ei - fh: ei = 1*0 = 0 and fh = 3*1 = 3, so ei - fh = 0 - 3 = -3.
Step 5: Calculate di - fg: di = 0*0 = 0 and fg = 3*2 = 6, so di - fg = 0 - 6 = -6.
Step 6: Calculate dh - eg: dh = 0*1 = 0 and eg = 1*2 = 2, so dh - eg = 0 - 2 = -2.
Step 7: Substitute these values into the determinant formula: det(J) = 1*(-3) - 2*(-6) + 1*(-2).
Step 8: Calculate each term: 1*(-3) = -3, -2*(-6) = 12, and 1*(-2) = -2.
Step 9: Add the results together: -3 + 12 - 2 = 7.
Step 10: The final result is det(J) = 7.

Determinants – Understanding how to calculate the determinant of a 3x3 matrix using the cofactor expansion method.
Matrix Operations – Knowledge of basic matrix operations and properties, particularly related to determinants.