Question

evaluate the following determinants:
matrix of [ad+bc bd-ac
ac-bd ad+bc]

Answer 1

Dear Student,

$$\begin{gathered} \left|\begin{array}{cc}\mathrm{ad}+\mathrm{bc}& \mathrm{bd}-\mathrm{ac}\\ \mathrm{ac}-\mathrm{bd}& \mathrm{ad}+\mathrm{bc}\end{array}\right| \\ =\left|\begin{array}{cc}\mathrm{ad}& \mathrm{bd}\\ \mathrm{ac}& \mathrm{ad}\end{array}\right|+\left|\begin{array}{cc}\mathrm{bc}& -\mathrm{ac}\\ -\mathrm{bd}& \mathrm{bc}\end{array}\right| \\ =\mathrm{ad}\left|\begin{array}{cc}\mathrm{d}& \mathrm{b}\\ \mathrm{c}& \mathrm{a}\end{array}\right|+\mathrm{bc}\left|\begin{array}{cc}\mathrm{c}& -\mathrm{a}\\ -\mathrm{d}& \mathrm{b}\end{array}\right| \\ =\mathrm{ad}\left(\mathrm{ad}-\mathrm{bc}\right)+\mathrm{bc}\left(\mathrm{bc}-\mathrm{ad}\right) \\ ={\mathrm{a}}^2{\mathrm{d}}^2-\mathrm{abcd}+{\mathrm{b}}^2{\mathrm{c}}^2-\mathrm{abcd} \\ ={\mathrm{a}}^2{\mathrm{d}}^2+{\mathrm{b}}^2{\mathrm{c}}^2-2\mathrm{abcd} \end{gathered}$$

Regards