# Determinant of $\left[\begin{array}{cc}e^{4 t} & e^{- \frac{7 t}{2}}\\4 e^{4 t} & - \frac{7 e^{- \frac{7 t}{2}}}{2}\end{array}\right]$

The calculator will find the determinant of the square $2$x$2$ matrix $\left[\begin{array}{cc}e^{4 t} & e^{- \frac{7 t}{2}}\\4 e^{4 t} & - \frac{7 e^{- \frac{7 t}{2}}}{2}\end{array}\right]$, with steps shown.

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### Your Input

Calculate $\left|\begin{array}{cc}e^{4 t} & e^{- \frac{7 t}{2}}\\4 e^{4 t} & - \frac{7 e^{- \frac{7 t}{2}}}{2}\end{array}\right|$.

### Solution

The determinant of a 2x2 matrix is $\left|\begin{array}{cc}a & b\\c & d\end{array}\right| = a d - b c$.

$\left|\begin{array}{cc}e^{4 t} & e^{- \frac{7 t}{2}}\\4 e^{4 t} & - \frac{7 e^{- \frac{7 t}{2}}}{2}\end{array}\right| = \left(e^{4 t}\right)\cdot \left(- \frac{7 e^{- \frac{7 t}{2}}}{2}\right) - \left(e^{- \frac{7 t}{2}}\right)\cdot \left(4 e^{4 t}\right) = - \frac{15 e^{\frac{t}{2}}}{2}$

### Answer

$\left|\begin{array}{cc}e^{4 t} & e^{- \frac{7 t}{2}}\\4 e^{4 t} & - \frac{7 e^{- \frac{7 t}{2}}}{2}\end{array}\right| = - \frac{15 e^{\frac{t}{2}}}{2} = - 7.5 e^{\frac{t}{2}}$A