Calculate the determinant of D = [[4, 5, 6], [7, 8, 9], [1, 2, 3]]. (2020)

Calculate the determinant of D = [[4, 5, 6], [7, 8, 9], [1, 2, 3]]. (2020)

Step 1: Identify the matrix D, which is D = [[4, 5, 6], [7, 8, 9], [1, 2, 3]].
Step 2: Use 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 = 4, b = 5, c = 6, d = 7, e = 8, f = 9, g = 1, h = 2, i = 3.
Step 4: Calculate ei - fh: ei = 8*3 = 24 and fh = 9*2 = 18, so ei - fh = 24 - 18 = 6.
Step 5: Calculate di - fg: di = 7*3 = 21 and fg = 9*1 = 9, so di - fg = 21 - 9 = 12.
Step 6: Calculate dh - eg: dh = 7*2 = 14 and eg = 8*1 = 8, so dh - eg = 14 - 8 = 6.
Step 7: Substitute these values into the determinant formula: det(D) = 4(6) - 5(12) + 6(6).
Step 8: Calculate each term: 4(6) = 24, -5(12) = -60, and 6(6) = 36.
Step 9: Combine the results: 24 - 60 + 36 = 0.
Step 10: Conclude that the determinant of D is 0.

Determinant Calculation – The process of calculating the determinant of a 3x3 matrix using the rule of Sarrus or cofactor expansion.
Matrix Properties – Understanding properties of determinants, such as how they relate to the invertibility of matrices.