Show that there is no $2 \times 2$ matrix $B$ such that $B^2 = A$

Question

$$A = \begin{bmatrix} 1 & 2\\ 3 & 4\end{bmatrix}$$

It’s been years since I took linear algebra and my niece just asked me for help with this problem. Can anyone help please? Thank you!

Answer 1

The following is under the assumption that you actually meant that there is no real matrix $\;B\;$ such that...

$$A=\begin{pmatrix}1&2\\3&4\end{pmatrix}=B^2\implies \det A=\det B^2=(\det B)^2$$

and now check what the determinant of $\;A\;$ is...