Calculate the determinant of F = [[1, 2, 3], [0, 1, 4], [5, 6, 0]]. (2023)

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Question: Calculate the determinant of F = [[1, 2, 3], [0, 1, 4], [5, 6, 0]]. (2023)

Options:

Correct Answer: -14

Exam Year: 2023

Solution:

Using the determinant formula, det(F) = 1(1*0 - 4*6) - 2(0*0 - 4*5) + 3(0*6 - 1*5) = -24 + 40 - 15 = 1.

Calculate the determinant of F = [[1, 2, 3], [0, 1, 4], [5, 6, 0]]. (2023)

Calculate the determinant of F = [[1, 2, 3], [0, 1, 4], [5, 6, 0]]. (2023)

Step 1: Identify the matrix F, which is F = [[1, 2, 3], [0, 1, 4], [5, 6, 0]].
Step 2: Use the determinant formula for a 3x3 matrix: det(F) = a(ei - fh) - b(di - fg) + c(dh - eg), where F = [[a, b, c], [d, e, f], [g, h, i]].
Step 3: Assign values from the matrix to the variables: a = 1, b = 2, c = 3, d = 0, e = 1, f = 4, g = 5, h = 6, i = 0.
Step 4: Calculate ei - fh: ei = 1*0 = 0 and fh = 4*6 = 24, so ei - fh = 0 - 24 = -24.
Step 5: Calculate di - fg: di = 0*0 = 0 and fg = 4*5 = 20, so di - fg = 0 - 20 = -20.
Step 6: Calculate dh - eg: dh = 0*6 = 0 and eg = 1*5 = 5, so dh - eg = 0 - 5 = -5.
Step 7: Substitute these values into the determinant formula: det(F) = 1*(-24) - 2*(-20) + 3*(-5).
Step 8: Calculate each term: 1*(-24) = -24, -2*(-20) = 40, and 3*(-5) = -15.
Step 9: Add these results together: -24 + 40 - 15 = 1.
Step 10: The determinant of matrix F is 1.

Determinant Calculation – The process of calculating the determinant of a 3x3 matrix using the cofactor expansion method.
Cofactor Expansion – Using the elements of a row or column to expand the determinant into smaller determinants.
Matrix Properties – Understanding how the arrangement of elements in a matrix affects the determinant value.