Calculate the determinant of D = [[3, 2, 1], [1, 0, 2], [2, 1, 3]]. (2020)

Calculate the determinant of D = [[3, 2, 1], [1, 0, 2], [2, 1, 3]]. (2020)

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

Determinant Calculation – The process of calculating the determinant of a 3x3 matrix using the formula involving minors and cofactors.
Matrix Properties – Understanding how the determinant relates to the properties of the matrix, such as invertibility and volume scaling.