Let $A,B\in M_{2\times2}\left(\mathbb{F}\right)$ such that $\det A\neq0$ and $\det B=2$. Calculate $\det\left(5AB^{3}A^{-1}B^{-1}\right)$.

Question

Let $A,B\in M_{2\times2}\left(\mathbb{F}\right)$ such that $\det A\neq0$ and $\det B=2$. Calculate $\det\left(5AB^{3}A^{-1}B^{-1}\right)$.

solution.

$\det\left(5AB^{3}A^{-1}B^{-1}\right)=5\cdot\det A\cdot\left(\det B\right)^{3}\cdot\frac{1}{\det A}\cdot\frac{1}{\det B}=5\cdot2^{3}\cdot\frac{1}{2}=5\cdot4=20$

Is this right?

Answer 1

You use the fact that $\det(MN)=\det(M)\det(N)$, which is right. However, for 2x2 matrices, $\det(5M)\neq 5\det(M)$. Indeed, $\det(5M)=\det((5I_2)M)=\det(5I_2)\det(M)=25\det(N)$.