Question

determine o determinante da matriz senx cosx -2cosx 2senx

Answer 1

$\boxed{\boxed{\left|\begin{array}{cc}a&b\\c&d\end{array}\right|=ad-bc}}$
_________________________________

$\left|\begin{array}{cc}sen(x)&cos(x)\\-2cos(x)&2sen(x)\end{array}\right|=sen(x)\cdot2sen(x)-cos(x)\cdot(-2)cos(x)\\\\\\\left|\begin{array}{cc}sen(x)&cos(x)\\-2cos(x)&2sen(x)\end{array}\right|=2sen^{2}(x)+2cos^{2}(x)$

Colocando 2 em evidência:

$\left|\begin{array}{cc}sen(x)&cos(x)\\-2cos(x)&2sen(x)\end{array}\right|=2\cdot(sen^{2}(x)+cos^{2}(x))$

Sabemos, pela relação fundamental da trigonometria, que sen²(x) + cos²(x) = 1:

$\left|\begin{array}{cc}sen(x)&cos(x)\\-2cos(x)&2sen(x)\end{array}\right|=2\cdot1\\\\\\\boxed{\boxed{\left|\begin{array}{cc}sen(x)&cos(x)\\-2cos(x)&2sen(x)\end{array}\right|=2}}$