Find the value of the determinant: | 2 3 1 | | 1 0 4 | | 5 6 7 |

Find the value of the determinant: | 2 3 1 | | 1 0 4 | | 5 6 7 |

Correct Answer: -20

Step 1: Write down the matrix for which we want to find the determinant: | 2 3 1 | | 1 0 4 | | 5 6 7 |.
Step 2: Use the formula for the determinant of a 3x3 matrix: | a b c | | d e f | | g h i | = a(ei - fh) - b(di - fg) + c(dh - eg).
Step 3: Identify the values from the matrix: a = 2, b = 3, c = 1, d = 1, e = 0, f = 4, g = 5, h = 6, i = 7.
Step 4: Calculate ei - fh: 0*7 - 4*6 = 0 - 24 = -24.
Step 5: Calculate di - fg: 1*7 - 4*5 = 7 - 20 = -13.
Step 6: Calculate dh - eg: 1*6 - 0*5 = 6 - 0 = 6.
Step 7: Substitute these values into the determinant formula: 2*(-24) - 3*(-13) + 1*(6).
Step 8: Calculate each term: 2*(-24) = -48, -3*(-13) = 39, 1*(6) = 6.
Step 9: Add these results together: -48 + 39 + 6 = -3.
Step 10: The final value of the determinant is -3.

Determinants – The determinant is a scalar value that can be computed from the elements of a square matrix and provides important properties about the matrix, such as whether it is invertible.
Matrix Operations – Understanding how to perform operations on matrices, including addition, multiplication, and finding determinants.