Evaluate the determinant \( \begin{vmatrix} 1 & 2 & 3 \\ 0 & 1 &

Evaluate the determinant \( \begin{vmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{vmatrix} \).

Evaluate the determinant \( \begin{vmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{vmatrix} \).

Step 1: Write down the determinant you want to evaluate: | 1 2 3 | | 0 1 4 | | 5 6 0 |.
Step 2: Use the formula for a 3x3 determinant: | a b c | = a(ei - fh) - b(di - fg) + c(dh - eg).
Step 3: Identify the elements: a = 1, b = 2, c = 3, d = 0, e = 1, f = 4, g = 5, h = 6, i = 0.
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: 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 -12.

Determinants – The question tests the ability to calculate the determinant of a 3x3 matrix using cofactor expansion or other methods.
Matrix Operations – Understanding how to manipulate and evaluate matrices is essential for solving the problem.