If H = [[1, 2, 3], [0, 1, 4], [5, 6, 0]], find the determinant of H. (2022)

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Question: If H = [[1, 2, 3], [0, 1, 4], [5, 6, 0]], find the determinant of H. (2022)

Options:

Correct Answer: -24

Exam Year: 2022

Solution:

Using the determinant formula for 3x3 matrices, det(H) = 1(1*0 - 4*6) - 2(0 - 4*5) + 3(0 - 1*5) = 0 - 40 - 15 = -55.

If H = [[1, 2, 3], [0, 1, 4], [5, 6, 0]], find the determinant of H. (2022)

If H = [[1, 2, 3], [0, 1, 4], [5, 6, 0]], find the determinant of H. (2022)

Step 1: Identify the matrix H, which is H = [[1, 2, 3], [0, 1, 4], [5, 6, 0]].
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 the matrix is [[a, b, c], [d, e, f], [g, h, i]].
Step 3: Assign the values from matrix H to the variables: a = 1, b = 2, c = 3, d = 0, e = 1, f = 4, g = 5, h = 6, i = 0.
Step 4: Calculate ei - fh: ei = 1*0 = 0 and fh = 4*6 = 24, so ei - fh = 0 - 24 = -24.
Step 5: Calculate di - fg: di = 0*0 = 0 and fg = 4*5 = 20, so di - fg = 0 - 20 = -20.
Step 6: Calculate dh - eg: dh = 0*6 = 0 and eg = 1*5 = 5, so dh - eg = 0 - 5 = -5.
Step 7: Substitute these values back into the determinant formula: det(H) = 1*(-24) - 2*(-20) + 3*(-5).
Step 8: Calculate each term: 1*(-24) = -24, -2*(-20) = 40, and 3*(-5) = -15.
Step 9: Combine the results: -24 + 40 - 15 = 1.
Step 10: The final result is det(H) = 1.

Determinant of a 3x3 Matrix – The determinant of a 3x3 matrix is calculated using the formula: det(H) = a(ei - fh) - b(di - eg) + c(dh - eg), where the matrix is represented as [[a, b, c], [d, e, f], [g, h, i]].