If H = [[1, 2, 1], [0, 1, 0], [2, 1, 1]], find det(H). (2021)

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Question: If H = [[1, 2, 1], [0, 1, 0], [2, 1, 1]], find det(H). (2021)

Options:

Correct Answer: 1

Exam Year: 2021

Solution:

The determinant of H is calculated as 1(1*1 - 0*1) - 2(0*1 - 0*2) + 1(0*1 - 1*2) = 1 - 0 - 2 = -1.

If H = [[1, 2, 1], [0, 1, 0], [2, 1, 1]], find det(H). (2021)

If H = [[1, 2, 1], [0, 1, 0], [2, 1, 1]], find det(H). (2021)

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

Determinant Calculation – The process of calculating the determinant of a 3x3 matrix using the formula involving the elements of the matrix and their respective minors.
Matrix Operations – Understanding how to perform operations on matrices, including multiplication and addition, which are foundational for determinant calculations.