Please help me with the linear algebra exercise below.
Let $A = \begin{pmatrix}a_{11} & a_{12}\\ a_{21} & a_{22}\end{pmatrix} \in \mathbb{C}^{2\times2}$ and $B\in\mathbb{C}^{n\times n}$. Denote by $A\otimes B\in\mathbb{C}^{2n\times 2n}$ the $2n\times 2n$ matrix with the block decomposition $$A\otimes B = \begin{pmatrix} a_{11}B & a_{12}B\\ a_{21}B & a_{22}B\end{pmatrix}.$$
(a) Assume that $a_{11}\neq0$ and that $B$ is invertible. Show that $\mbox{det}A\otimes B = (\mbox{det}A\cdot\mbox{det}B)^2$.
(b) Assume only that $B$ is invertible. Show that the formula $\mbox{det}A\otimes B = (\mbox{det}A\cdot\mbox{det}B)^2$ remains true when $a_{11}=0$. (Hint: Use a continuity argument.)
(c) Show that the formula $\mbox{det}A\otimes B = (\mbox{det}A\cdot\mbox{det}B)^2$ remains true even when $B$ is not invertible (and hence is always true). (Hint: Use the fact that the spectrum of $B$ consists of a finite number of points and use another continuity argument.)
(a) Remark that we can decompose $A\otimes B$ into product of two matrices as follows $$A\otimes B = \begin{pmatrix}I & 0\\ a_{21}B\cdot a_{11}^{-1}B & I\end{pmatrix}\cdot\begin{pmatrix}a_{11}B & a_{12}B\\ 0 & a_{22}B-a_{21}B\cdot a_{11}^{-1}B^{-1}\cdot a_{12}B\end{pmatrix},$$ hence $$\mbox{det}A\otimes B = a_{11}\mbox{det}B\cdot(a_{22}\mbox{det}B-a_{21}\mbox{det}B\cdot a_{11}^{-1}\mbox{det}B^{-1}\cdot a_{12}\mbox{det}B) = \mbox{det}A\cdot \mbox{det}B^2.$$
I can't see why $\mbox{det}A$ should be in the power of two here. Am I wrong or is there a typo in the problem?
(b) In fact I don't clearly understand the hint about "continuity argument". I know that determinant is a continuous function from matrix elements, so maybe I should consider $a_{11}-\varepsilon < a_{11} < a_{11} + \varepsilon$ with $\varepsilon\to0$, but, all in all, I didn't get it completely.
(c) The same trouble with "continuity argument". The hint part about spectrum is more or less clear since matrix determinant can be expressed in terms of matrix eigenvalues: $\mbox{det}B = \prod_{i=1}^n \lambda_i$, where $\lambda_i$ is an eigenvalue of $B$.
A general strategy for these continuity arguments is the following: if $\det(A_k \otimes B_k) = (\det A_k)^n(\det B_k)^2$ for matrices $A_k \in \mathbb{C}^{2 \times 2}$ and $B_k \in \mathbb{C}^{n \times n}$, and the sequences converge to $\lim_k A_k = A$ and $\lim_k B_k = B$ respectively, then $\det(A \otimes B) = (\det A)^n(\det B)^2$ as well.
For example, for part b) with general $A \in \mathbb{C}^{2\times 2}$, you need to find a sequence of matrices in $\mathbb{C}^{2 \times 2}$ converging to $A$ for which the identity holds.