<s>[INST] Determinant Question. Show that if $A=\begin{bmatrix}a & b\\c & d\end{bmatrix}$, then 
$\det(A)=\frac{1}{2}\det\left(\begin{bmatrix}1 & 1\\tr(A^2) & (tr(A))^2\end{bmatrix}\right)$.
I tried finding the determinant using the formula but it didn't come out to be the same as when I found determinant using cofactor expansion.
 [/INST] We have that 
$$det(A) = ad-bc$$
$$A^{2} = \begin{bmatrix} a^{2} + bc & ab + bd \\ ca + dc & cb + d^{2}\end{bmatrix}$$
Then 
\begin{align*}
\frac{1}{2} \cdot det(\begin{bmatrix}1 & 1\\tr(A^2) & (tr(A))^2\end{bmatrix}) &=\frac{1}{2} \cdot det(\begin{bmatrix}1 & 1\\a^{2} + 2bc + d^{2} & (a+d)^{2}\end{bmatrix}) \\
&= \frac{1}{2}(a^{2} + 2ad + d^{2} - a^{2} - 2bc - d^{2}) \\
&= ad - bc 
\end{align*}
as required. 
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