Calculate the determinant of G = [[1, 2, 1], [0, 1, 0], [2, 3, 1]]. (2023)

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Question: Calculate the determinant of G = [[1, 2, 1], [0, 1, 0], [2, 3, 1]]. (2023)

Options:

Correct Answer: -1

Exam Year: 2023

Solution:

Using cofactor expansion, det(G) = 1(1*1 - 0*3) - 2(0*1 - 0*2) + 1(0*3 - 1*2) = 1 - 0 - 2 = -1.

Calculate the determinant of G = [[1, 2, 1], [0, 1, 0], [2, 3, 1]]. (2023)

Calculate the determinant of G = [[1, 2, 1], [0, 1, 0], [2, 3, 1]]. (2023)

Step 1: Identify the matrix G, which is G = [[1, 2, 1], [0, 1, 0], [2, 3, 1]].
Step 2: Choose the first row for cofactor expansion. The first row is [1, 2, 1].
Step 3: Calculate the determinant using the formula: det(G) = a11 * det(M11) - a12 * det(M12) + a13 * det(M13), where aij is the element in the ith row and jth column, and Mij is the minor matrix after removing the ith row and jth column.
Step 4: For a11 = 1, remove the first row and first column to get M11 = [[1, 0], [3, 1]]. Calculate det(M11) = (1*1) - (0*3) = 1.
Step 5: For a12 = 2, remove the first row and second column to get M12 = [[0, 0], [2, 1]]. Calculate det(M12) = (0*1) - (0*2) = 0.
Step 6: For a13 = 1, remove the first row and third column to get M13 = [[0, 1], [2, 3]]. Calculate det(M13) = (0*3) - (1*2) = -2.
Step 7: Substitute the values back into the determinant formula: det(G) = 1(1) - 2(0) + 1(-2).
Step 8: Simplify the expression: det(G) = 1 - 0 - 2 = -1.

Determinant Calculation – The process of finding the determinant of a matrix using cofactor expansion.
Cofactor Expansion – A method for calculating the determinant by expanding along a row or column.
Matrix Properties – Understanding how the elements of a matrix interact in determinant calculations.