If A is a square matrix of order 2 and det. A = 10, then ((tr. A)² - tr.(A²)) is equal to ___________.

Let $\mathrm{A}=\left[\begin{array}{l}\mathrm{a}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{b}\\ \mathrm{c}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{d}\end{array}\right]$

$\begin{array}{c}\therefore {\mathrm{A}}^2=\left[\begin{array}{ll}\mathrm{a}& \mathrm{b}\\ \mathrm{c}& \mathrm{d}\end{array}\right]\left[\begin{array}{ll}\mathrm{a}& \mathrm{b}\\ \mathrm{c}& \mathrm{d}\end{array}\right]\\ =\left[\begin{array}{ll}{\mathrm{a}}^2+\mathrm{bc}& \mathrm{ab}+\mathrm{bd}\\ \mathrm{ac}+\mathrm{cd}& \mathrm{bc}+{\mathrm{d}}^2\end{array}\right]\end{array}$

$\begin{array}{l}\therefore (\text{ tr. }\mathrm{A}{)}^2-\left(\text{ tr. }{\mathrm{A}}^2\right)=(\mathrm{a}+\mathrm{d}{)}^2-\left({\mathrm{a}}^2+\mathrm{bc}+\mathrm{bc}+{\mathrm{d}}^2\right)\\ =2(\mathrm{ad}-\mathrm{bc})=2\text{ det. }\mathrm{A}=20\end{array}$