Introduction:
We are given a 3x3 Hermitian matrix A that satisfies the matrix equation M² - 7M + 12I = 0, where I refers to the identity matrix. We are asked to find the determinant of matrix M, given that the trace is 10.

Hermitian Matrix:
A Hermitian matrix is a square matrix that is equal to its own conjugate transpose. In other words, the complex conjugate of each element in the matrix is equal to its transpose. In this case, matrix A is a 3x3 Hermitian matrix.

Matrix Equation:
The given equation is M² - 7M + 12I = 0, where I represents the identity matrix. This equation can be rewritten as M² - 7M + 12I = 0I. Simplifying further, we have M(M - 7I) = -12I.

Trace:
The trace of a square matrix is the sum of the elements on its main diagonal. In this case, we know that the trace of matrix M is 10.

Determinant of Matrix M:
To find the determinant of matrix M, we can use the fact that the determinant of a product of matrices is equal to the product of their determinants. From the given equation M(M - 7I) = -12I, we can see that the determinant of M(M - 7I) is equal to the determinant of -12I.

Determinant of Identity Matrix:
The determinant of an identity matrix is always 1. In this case, the determinant of -12I is (-12)³ * det(I), which simplifies to -1728.

Determinant of Matrix M:
Using the fact that the determinant of a product of matrices is equal to the product of their determinants, we have det(M(M - 7I)) = det(M) * det(M - 7I) = -1728. Since we know that the trace of M is 10, we also know that the sum of the eigenvalues of M is 10.

Eigenvalues of Matrix M:
The eigenvalues of a matrix are the values λ for which the equation Mx = λx holds true, where x is a non-zero vector. In this case, we can denote the eigenvalues as λ₁, λ₂, and λ₃.

Sum of Eigenvalues:
The sum of the eigenvalues of a matrix is equal to its trace. In this case, we know that the sum of the eigenvalues of M is 10.

Product of Eigenvalues:
The product of the eigenvalues of a matrix is equal to its determinant. In this case, we have λ₁ * λ₂ * λ₃ = -1728.

Determinant of Matrix M:
From the above equation, we can see that the determinant of matrix M is -1728.

Conclusion:
The determinant of matrix M, given that the trace is 10, is -1728.