Calculate the determinant: | 2 3 1 | | 1 0 4 | | 0 5 2 |.

Calculate the determinant: | 2 3 1 | | 1 0 4 | | 0 5 2 |.

Correct Answer: 0

Step 1: Write down the matrix: | 2 3 1 | | 1 0 4 | | 0 5 2 |.
Step 2: Identify the elements of the matrix: a = 2, b = 3, c = 1, d = 1, e = 0, f = 4, g = 0, h = 5, i = 2.
Step 3: Use the determinant formula for a 3x3 matrix: det(A) = a(ei - fh) - b(di - fg) + c(dh - eg).
Step 4: Calculate ei - fh: (0*2) - (4*5) = 0 - 20 = -20.
Step 5: Calculate di - fg: (1*2) - (4*0) = 2 - 0 = 2.
Step 6: Calculate dh - eg: (1*5) - (0*0) = 5 - 0 = 5.
Step 7: Substitute these values into the determinant formula: det(A) = 2*(-20) - 3*(2) + 1*(5).
Step 8: Calculate: det(A) = -40 - 6 + 5 = -41.
Step 9: Since the determinant is not zero, the final answer is that the determinant evaluates to -41.

Determinant Calculation – The process of calculating the determinant of a 3x3 matrix using the formula involving the elements of the matrix.
Properties of Determinants – Understanding that a determinant can be zero if the rows or columns of the matrix are linearly dependent.