For the matrix J = [[1, 2, 3], [0, 1, 4], [5, 6, 0]], find the determinant. (202

For the matrix J = [[1, 2, 3], [0, 1, 4], [5, 6, 0]], find the determinant. (2023)

For the matrix J = [[1, 2, 3], [0, 1, 4], [5, 6, 0]], find the determinant. (2023)

Step 1: Identify the matrix J, which is J = [[1, 2, 3], [0, 1, 4], [5, 6, 0]].
Step 2: Use the determinant formula for a 3x3 matrix: det(J) = a(ei - fh) - b(di - eg) + c(dh - eg), where the matrix is structured as follows: [[a, b, c], [d, e, f], [g, h, i]].
Step 3: Assign values from the matrix to the variables: 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 - eg = (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: Calculate each term: 1*(-24) = -24, -2*(-20) = 40, 3*(-5) = -15.
Step 9: Combine the results: -24 + 40 - 15 = 1.
Step 10: The final result is det(J) = 1.

Determinant Calculation – The process of calculating the determinant of a 3x3 matrix using the cofactor expansion method.
Matrix Operations – Understanding how to perform basic operations with matrices, including multiplication and addition, which are often involved in determinant calculations.