Calculate the determinant of D = [[3, 2, 1], [1, 0, 2], [0, 1, 3]]. (2023)

Calculate the determinant of D = [[3, 2, 1], [1, 0, 2], [0, 1, 3]]. (2023)

Step 1: Identify the matrix D = [[3, 2, 1], [1, 0, 2], [0, 1, 3]].
Step 2: Write down the formula for the determinant of a 3x3 matrix: Det(D) = 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 the matrix to the variables: a = 3, b = 2, c = 1, d = 1, e = 0, f = 2, g = 0, h = 1, i = 3.
Step 4: Calculate the first part: ei - fh = (0*3) - (2*1) = 0 - 2 = -2.
Step 5: Calculate the second part: di - fg = (1*3) - (2*0) = 3 - 0 = 3.
Step 6: Calculate the third part: dh - eg = (1*1) - (0*0) = 1 - 0 = 1.
Step 7: Substitute these values back into the determinant formula: Det(D) = 3(-2) - 2(3) + 1(1).
Step 8: Calculate each term: 3(-2) = -6, -2(3) = -6, and 1(1) = 1.
Step 9: Combine the results: -6 - 6 + 1 = -12 + 1 = -11.
Step 10: The final result is Det(D) = -11.

Determinant Calculation – The process of calculating the determinant of a 3x3 matrix using the rule of Sarrus or cofactor expansion.
Cofactor Expansion – Using the elements of a row or column and their corresponding minors to compute the determinant.