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

₹0.0

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

Options:

Correct Answer: -4

Exam Year: 2022

Solution:

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

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

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

Step 1: Write down the matrix F: F = [[2, 1, 3], [1, 0, 2], [3, 1, 1]].
Step 2: Identify the elements of the matrix: a = 2, b = 1, c = 3, d = 1, e = 0, f = 2, g = 3, h = 1, 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*1) - 1(1*1 - 2*3) + 3(1*1 - 0*3).
Step 5: Calculate each part: 0*1 = 0, 2*1 = 2, so ei - fh = 0 - 2 = -2; 1*1 = 1, 2*3 = 6, so di - fg = 1 - 6 = -5; 1*1 = 1, 0*3 = 0, so dh - eg = 1 - 0 = 1.
Step 6: Substitute these results back into the equation: det(F) = 2(-2) - 1(-5) + 3(1).
Step 7: Calculate: 2*(-2) = -4, -1*(-5) = 5, 3*1 = 3.
Step 8: Add these results together: -4 + 5 + 3 = 4.
Step 9: The determinant of matrix F is 4.

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 Convention – Recognizing the importance of the sign in the determinant calculation, especially when applying the cofactor expansion.