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

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

Solution: Det(D) = 3(0*3 - 2*1) - 2(1*3 - 0*2) + 1(1*1 - 0*0) = 3(0 - 2) - 2(3) + 1(1) = -6 - 6 + 1 = -11.

Steps: 10

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.