Determinant of $M = \begin{pmatrix} I_n&iI_n \\iI_n&I_n \end{pmatrix}$

Question

Calculate the determinant of the following matrix:

$M \in M_{2n}(\mathbb{C})$ such that $$M = \begin{pmatrix} I_n&iI_n \\iI_n&I_n \end{pmatrix}$$

I find that that $\det M = 2^n$ is that correct and if not please provide some details.

Answer 1

There exists a generalization of Cofactor expansion called Laplace expansion. It is a very cumbersome method but it has a very natural and useful special case. Let $$\begin{pmatrix}A & 0 \\ B & C\end{pmatrix}$$ be a matrix composed of block-matrices $A,B,C,0$ of appropriate dimensions so that this matrix makes sense. The matrix $0$ is a is a zero matrix. Then this result says $$\det \begin{pmatrix}A & 0 \\ B & C\end{pmatrix} = \det (A) \det (C)$$ i.e that the method for expanding upper triangular matrizcs actually also extends to 'upper-triangular-block-matrices'.

For your matrix we can add $-i$ times the second 'block-row' to the first block-row to get $$\det M = \det \begin{pmatrix}2 I_n & 0 \\ iI_n & I_n \end{pmatrix} = \det(2I_n) \det(I_n) = 2^n$$