If F = [[2, 1, 3], [1, 0, 2], [0, 1, 1]], what is det(F)? (2023)

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Question: If F = [[2, 1, 3], [1, 0, 2], [0, 1, 1]], what is det(F)? (2023)

Options:

Correct Answer: 3

Exam Year: 2023

Solution:

Det(F) = 2(0*1 - 2*1) - 1(1*1 - 2*0) + 3(1*1 - 0*0) = 2(0 - 2) - 1(1) + 3(1) = -4 - 1 + 3 = -2.

If F = [[2, 1, 3], [1, 0, 2], [0, 1, 1]], what is det(F)? (2023)

If F = [[2, 1, 3], [1, 0, 2], [0, 1, 1]], what is det(F)? (2023)

Step 1: Identify the matrix F. F = [[2, 1, 3], [1, 0, 2], [0, 1, 1]].
Step 2: Write down the formula for the determinant of 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 F to the variables: a = 2, b = 1, c = 3, d = 1, e = 0, f = 2, g = 0, h = 1, i = 1.
Step 4: Calculate the first part: ei - fh = (0*1) - (2*1) = 0 - 2 = -2.
Step 5: Calculate the second part: di - fg = (1*1) - (2*0) = 1 - 0 = 1.
Step 6: Calculate the third part: dh - eg = (1*1) - (0*0) = 1 - 0 = 1.
Step 7: Substitute these values back into the determinant formula: det(F) = 2(-2) - 1(1) + 3(1).
Step 8: Calculate each term: 2(-2) = -4, -1(1) = -1, and 3(1) = 3.
Step 9: Combine the results: -4 - 1 + 3 = -2.
Step 10: Conclude that det(F) = -2.

Determinants – The determinant of a matrix is a scalar value that can be computed from its elements and provides important properties about the matrix, such as whether it is invertible.
Cofactor Expansion – The method used to calculate the determinant by expanding along a row or column, involving minors and cofactors.
Matrix Operations – Understanding how to perform basic operations on matrices, including multiplication and addition, which are often involved in determinant calculations.