Find the value of the determinant: | 2 3 1 | | 1 0 4 | | 5 6 2 |

Find the value of the determinant: | 2 3 1 | | 1 0 4 | | 5 6 2 |

Step 1: Write down the matrix for which we want to find the determinant: | 2 3 1 | | 1 0 4 | | 5 6 2 |.
Step 2: Identify the elements of the matrix: a = 2, b = 3, c = 1, d = 1, e = 0, f = 4, g = 5, h = 6, 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*6) = 0 - 24 = -24.
Step 5: Calculate di - fg: (1*2) - (4*5) = 2 - 20 = -18.
Step 6: Calculate dh - eg: (1*6) - (0*5) = 6 - 0 = 6.
Step 7: Substitute these values into the determinant formula: det(A) = 2*(-24) - 3*(-18) + 1*(6).
Step 8: Calculate each term: 2*(-24) = -48, -3*(-18) = 54, 1*(6) = 6.
Step 9: Add these results together: -48 + 54 + 6 = 12.
Step 10: The final value of the determinant is -10.

Determinant Calculation – The process of finding the determinant of a 3x3 matrix using the formula involving the elements of the matrix.
Matrix Operations – Understanding how to manipulate and calculate properties of matrices, including determinants.