Yes, a matrix can have a complex determinant. In linear algebra, the determinant of a matrix is a scalar value that can be real, complex, or even zero. It represents certain properties of the matrix and is used in various calculations and theorems.

  A complex determinant occurs when the matrix contains complex numbers in its entries. Complex numbers are numbers that consist of a real part and an imaginary part, represented as a + bi, where a and b are real numbers and i is the imaginary unit.

  To calculate the complex determinant of a matrix, you can use methods such as expansion by cofactors, Gaussian elimination, or using the properties of determinants. These methods apply to matrices whose entries are complex numbers as well.

  For example, consider the following 2x2 matrix:

  | 1 + i 2 - i |

  | 3 - 2i 4 + 3i |

  To find its determinant, you can use the formula:

  det(A) = (1 + i)(4 + 3i) - (2 - i)(3 - 2i)

  Simplifying, you get:

  det(A) = 1 + 7i - 2i + 3 - 6i - (6 - 3i + 2i - 2)

  det(A) = -3 + 7i -5i + 2i - 2 + 3 - 6i

  det(A) = 0 - 2i - 2 - 6i

  det(A) = -2 - 8i

  Therefore, the determinant of this matrix is a complex number, -2 - 8i.

  So, in conclusion, matrices can have complex determinants, and their calculation follows the same principles as matrices with real determinants.