What is the determinant of J = [[1, 2, 3], [0, 1, 4], [5, 6, 0]]? (2014)

1 question

1 item

Q: What is the determinant of J = [[1, 2, 3], [0, 1, 4], [5, 6, 0]]? (2014)

Solution: Det(J) = 1(1*0 - 4*6) - 2(0*0 - 4*5) + 3(0*6 - 1*5) = 1(0 - 24) - 2(0 - 20) + 3(0 - 5) = -24 + 40 - 15 = 1.

Steps: 10

Step 1: Identify the matrix J. J = [[1, 2, 3], [0, 1, 4], [5, 6, 0]].
Step 2: Write down the formula for the determinant of a 3x3 matrix. For a matrix [[a, b, c], [d, e, f], [g, h, i]], the determinant is: Det = a(ei - fh) - b(di - fg) + c(dh - eg).
Step 3: Assign values from matrix J to the formula. Here, a = 1, b = 2, c = 3, d = 0, e = 1, f = 4, g = 5, h = 6, i = 0.
Step 4: Calculate the first part: ei - fh = 1*0 - 4*6 = 0 - 24 = -24.
Step 5: Calculate the second part: di - fg = 0*0 - 4*5 = 0 - 20 = -20.
Step 6: Calculate the third part: dh - eg = 0*6 - 1*5 = 0 - 5 = -5.
Step 7: Substitute these values back into the determinant formula: Det(J) = 1*(-24) - 2*(-20) + 3*(-5).
Step 8: Simplify the expression: Det(J) = -24 + 40 - 15.
Step 9: Calculate the final result: -24 + 40 = 16, then 16 - 15 = 1.
Step 10: Conclude that the determinant of J is 1.