If F = [[2, 1, 3], [1, 0, 2], [3, 4, 1]], find det(F). (2022)

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Question: If F = [[2, 1, 3], [1, 0, 2], [3, 4, 1]], find det(F). (2022)

Options:

Correct Answer: -10

Solution:

Using the determinant formula, det(F) = 2*(0*1 - 2*4) - 1*(1*1 - 2*3) + 3*(1*4 - 0*3) = 2*(-8) - 1*(-5) + 3*4 = -16 + 5 + 12 = 1.

If F = [[2, 1, 3], [1, 0, 2], [3, 4, 1]], find det(F). (2022)

If F = [[2, 1, 3], [1, 0, 2], [3, 4, 1]], find det(F). (2022)

Step 1: Write down the matrix F: F = [[2, 1, 3], [1, 0, 2], [3, 4, 1]].
Step 2: Identify the elements of the matrix: a = 2, b = 1, c = 3, d = 1, e = 0, f = 2, g = 3, h = 4, i = 1.
Step 3: Use the determinant formula for a 3x3 matrix: det(F) = a(ei - fh) - b(di - fg) + c(dh - eg).
Step 4: Substitute the values into the formula: det(F) = 2*(0*1 - 2*4) - 1*(1*1 - 2*3) + 3*(1*4 - 0*3).
Step 5: Calculate each part: 0*1 = 0, 2*4 = 8, so ei - fh = 0 - 8 = -8.
Step 6: Calculate the second part: 1*1 = 1, 2*3 = 6, so di - fg = 1 - 6 = -5.
Step 7: Calculate the third part: 1*4 = 4, 0*3 = 0, so dh - eg = 4 - 0 = 4.
Step 8: Now substitute back into the determinant formula: det(F) = 2*(-8) - 1*(-5) + 3*(4).
Step 9: Calculate each term: 2*(-8) = -16, -1*(-5) = 5, 3*(4) = 12.
Step 10: Add the results together: -16 + 5 + 12 = 1.
Step 11: Therefore, the determinant of F is 1.

Determinants – The question tests the ability to calculate the determinant of a 3x3 matrix using the determinant formula.
Matrix Operations – Understanding how to perform operations on matrix elements to compute the determinant.
Sign and Coefficients – Recognizing the importance of signs and coefficients in the determinant expansion.