Calculate the determinant of D = [[2, 3, 1], [1, 0, 2], [4, 1, 0]]. (2020)

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Question: Calculate the determinant of D = [[2, 3, 1], [1, 0, 2], [4, 1, 0]]. (2020)

Options:

Correct Answer: -10

Exam Year: 2020

Solution:

Det(D) = 2(0*0 - 2*1) - 3(1*0 - 2*4) + 1(1*1 - 0*4) = 2(0 - 2) - 3(0 - 8) + 1(1) = -4 + 24 + 1 = 21.

Calculate the determinant of D = [[2, 3, 1], [1, 0, 2], [4, 1, 0]]. (2020)

Calculate the determinant of D = [[2, 3, 1], [1, 0, 2], [4, 1, 0]]. (2020)

Step 1: Write down the matrix D: [[2, 3, 1], [1, 0, 2], [4, 1, 0]].
Step 2: Identify the elements of the matrix: a = 2, b = 3, c = 1, d = 1, e = 0, f = 2, g = 4, h = 1, i = 0.
Step 3: Use the formula for the determinant of a 3x3 matrix: Det(D) = a(ei - fh) - b(di - fg) + c(dh - eg).
Step 4: Substitute the values into the formula: Det(D) = 2(0*0 - 2*1) - 3(1*0 - 2*4) + 1(1*1 - 0*4).
Step 5: Calculate each part: 0*0 = 0, 2*1 = 2, so ei - fh = 0 - 2 = -2.
Step 6: Calculate the second part: 1*0 = 0, 2*4 = 8, so di - fg = 0 - 8 = -8.
Step 7: Calculate the third part: 1*1 = 1, 0*4 = 0, so dh - eg = 1 - 0 = 1.
Step 8: Now substitute back into the determinant calculation: Det(D) = 2(-2) - 3(-8) + 1(1).
Step 9: Calculate: 2*(-2) = -4, -3*(-8) = 24, and 1(1) = 1.
Step 10: Add these results together: -4 + 24 + 1 = 21.
Step 11: The final result is Det(D) = 21.

Determinant Calculation – The process of calculating the determinant of a 3x3 matrix using the formula involving minors and cofactors.
Matrix Operations – Understanding how to perform operations on matrices, including multiplication and addition, which are often involved in determinant calculations.