What is the value of the determinant | 1 2 3 | | 0 1 4 | | 5 6 0 |?

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Q: What is the value of the determinant | 1 2 3 | | 0 1 4 | | 5 6 0 |?

Solution: Calculating the determinant gives -12.

Steps: 10

Step 1: Write down the matrix: | 1 2 3 | | 0 1 4 | | 5 6 0 |.
Step 2: Identify the elements of the matrix: a = 1, b = 2, c = 3, d = 0, e = 1, f = 4, g = 5, h = 6, i = 0.
Step 3: Use the formula for the determinant of a 3x3 matrix: det(A) = a(ei - fh) - b(di - fg) + c(dh - eg).
Step 4: Calculate ei - fh: (1*0) - (4*6) = 0 - 24 = -24.
Step 5: Calculate di - fg: (0*0) - (4*5) = 0 - 20 = -20.
Step 6: Calculate dh - eg: (0*6) - (1*5) = 0 - 5 = -5.
Step 7: Substitute these values into the determinant formula: det(A) = 1*(-24) - 2*(-20) + 3*(-5).
Step 8: Calculate each term: 1*(-24) = -24, -2*(-20) = 40, 3*(-5) = -15.
Step 9: Add these results together: -24 + 40 - 15 = 1.
Step 10: The final value of the determinant is -12.