Past exam of the mathematics course of the University of Cambridge / 2016 / ib / Paper 2 / 10F / b

Table of contents
- Solution b

Solution

Right multiplication by

B^{*}

acts separately on the rows. After transposing each row its matrix is

(B^{*})^{T} = \overline{B}

, so its determinant is

(\overline{det B})^{n}

. Composing this linear map with left multiplication and using part (a) gives

det M_{B} = (det B)^{n} (\overline{det B})^{n} = ∣ det B ∣^{2 n} .

Here the determinant is taken on the

n^{2}

-dimensional complex vector space of matrices.

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