Exercise 10.B.8

Proof. Suppose $𝔽=ℂ$. Choose an orthonormal basis of $V$ with respect to which the matrix of $T$ is upper triangular. Then, by 10.35, the determinant of $M(T)$ equals the product of the entries on the diagonal. By the same reasoning used in 10.35, the determinant of $M(T^{\text{∗}})$ is also the product of the entries on diagonal. The diagonal entries of $M(T^{\text{∗}})$ are the conjugates of the diagonal entries of $M(T)$. Hence $\det <!--\; FUNCTION\; APPLICATION\; -->M(T^{\text{∗}})=\overline{\det <!--\; FUNCTION\; APPLICATION\; -->M(T)}$. Now 10.42 implies that $\det <!--\; FUNCTION\; APPLICATION\; -->T^{\text{∗}}=\overline{\det <!--\; FUNCTION\; APPLICATION\; -->T}$.

Form 10.44, we have

Taking the square root of each side we get $\det <!--\; FUNCTION\; APPLICATION\; -->\sqrt{T^{\ast }T}=\left|\det <!--\; FUNCTION\; APPLICATION\; -->T\right|$. □