Calculate the determinant of G = [[1, 2, 1], [0, 1, 4], [1, 0, 0]]. (2021)

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Question: Calculate the determinant of G = [[1, 2, 1], [0, 1, 4], [1, 0, 0]]. (2021)

Options:

Correct Answer: -2

Exam Year: 2021

Solution:

Using the determinant formula for 3x3 matrices, det(G) = 1(1*0 - 4*0) - 2(0*0 - 4*1) + 1(0*0 - 1*1) = 0 + 8 - 1 = 7.

Calculate the determinant of G = [[1, 2, 1], [0, 1, 4], [1, 0, 0]]. (2021)

Calculate the determinant of G = [[1, 2, 1], [0, 1, 4], [1, 0, 0]]. (2021)

Step 1: Identify the matrix G. G = [[1, 2, 1], [0, 1, 4], [1, 0, 0]].
Step 2: Write down the formula for the determinant of a 3x3 matrix: det(G) = a(ei - fh) - b(di - fg) + c(dh - eg), where G = [[a, b, c], [d, e, f], [g, h, i]].
Step 3: Assign values from matrix G to the variables: a = 1, b = 2, c = 1, d = 0, e = 1, f = 4, g = 1, h = 0, i = 0.
Step 4: Calculate the first part: ei - fh = (1*0) - (4*0) = 0.
Step 5: Calculate the second part: di - fg = (0*0) - (4*1) = 0 - 4 = -4.
Step 6: Calculate the third part: dh - eg = (0*0) - (1*1) = 0 - 1 = -1.
Step 7: Substitute these values back into the determinant formula: det(G) = 1(0) - 2(-4) + 1(-1).
Step 8: Simplify the expression: det(G) = 0 + 8 - 1.
Step 9: Final calculation: det(G) = 7.

Determinant of a 3x3 Matrix – The determinant of a 3x3 matrix can be calculated using the formula: det(A) = a(ei - fh) - b(di - fg) + c(dh - eg), where A = [[a, b, c], [d, e, f], [g, h, i]].
Matrix Multiplication and Addition – Understanding how to perform multiplication and addition of matrix elements is crucial for calculating determinants.