Previous in Matrices and Determinants Next in Matrices and Determinants
Question Number 149116 by bramlexs22 last updated on 03/Aug/21
$$\:\mathrm{If}\:\mathrm{A}\:\mathrm{is}\:\mathrm{a}\:\mathrm{3}×\mathrm{3}\:\mathrm{matrix}\:\mathrm{where}\:\mathrm{det}\left(\mathrm{A}\right)=−\mathrm{2} \\ $$$$\mathrm{then}\:\mathrm{what}\:\mathrm{will}\:\mathrm{be}\:\mathrm{det}\left(\mathrm{3A}^{\mathrm{2}} \mathrm{A}^{−\mathrm{1}} \right)? \\ $$$$\mathrm{knowing}\:\mathrm{that}\:\mathrm{A}^{−\mathrm{1}} \:\mathrm{is}\:\mathrm{the}\:\mathrm{inverse} \\ $$$$\mathrm{of}\:\mathrm{A}\: \\ $$
Answered by EDWIN88 last updated on 03/Aug/21
$${det}\left(\mathrm{3}{A}^{\mathrm{2}} {A}^{−\mathrm{1}} \right)={det}\left(\mathrm{3}{A}\right)=\mathrm{3}^{\mathrm{3}} ×{det}\left({A}\right) \\ $$$$=\mathrm{27}×\left(−\mathrm{2}\right)=−\mathrm{54} \\ $$