In (9.1) we have defined the adjoint of a matrix as the transpose conjugate. This is the usual definition except in algebra where the adjoint is defined as the transposed matrix of cofactors [see (6.13)]. Show that the two definitions are the same for a unitary matrix with determinant$=+1$

Determinant of Hermitian Conjugate matrix.

The determinant is a scalar value that functions as a function of the square matrix entries. It allows characterisation of some of the matrix's attributes as well as the linear map that the matrix represents.

The adjugate or classical adjoint of a square matrix is the transpose of its cofactor matrix in linear algebra.

In linear algebra, there are two ways to define a Hermitian conjugate of a matrix, one being a transpose and complex conjugate of a matrix, and the other a transpose matrix of cofactors. For a unitary matrix, we have

${\mathrm{A}}^{-1}={\mathrm{A}}^{†}$

The former relation gives

$\frac{1}{\det A}{C}^{\mathrm{\top }}=A^{†}$

If the determinant of A is one, we have

$C^{\mathrm{\top }}=A^{†}$

Which says that the Hermitian conjugate can be defined as a transpose cofactor matrix