Final answer:
To check if matrix A is unitary, one must multiply matrix A with its conjugate transpose A* and see if the result is the identity matrix. Furthermore, the determinant of matrix A can be found using the standard method of expansion of cofactors.
Step-by-step explanation:
The task is to verify whether the matrix A is unitary and to calculate its determinant. A matrix is unitary if the product of the matrix and its conjugate transpose results in the identity matrix. The conjugate transpose of matrix A, denoted as A*, is obtained by taking the transpose of the matrix and then taking the complex conjugate of each element. To verify A is unitary, we calculate AA* and check if it equals the identity matrix.
Verification of Unitary Matrix:
A = \begin{pmatrix}
1 & i & 2+3i \\
-i & 0 & 7 \\
2-3i & 7 & 4+i
\end{pmatrix}
A* (Conjugate Transpose of A) = \begin{pmatrix}
1 & -i & 2-3i \\
i & 0 & 7 \\
2+3i & 7 & 4-i
\end{pmatrix}
To verify, we need to calculate the product AA* and check for the identity matrix.
Calculation of Determinant:
The determinant of a matrix can be calculated using the formula provided in Equation 2.32 or by expansion of cofactors. We follow the latter method for this complex matrix to find the determinant of A.