Question

Find the value of each determinant.

19.$\left|\begin{array}{cc}2& -7\\ -5& 3\end{array}\right|$

Short answer

The value of the determinant $\left|\begin{array}{cc}2& -7\\ -5& 3\end{array}\right|=-29$.

Step-by-step solution

- Define the determinant of second-order matrix

The determinant of second order matrix is found by calculating the difference of the product of the two diagonals, that is., $\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|=ad-bc$.

- Find the determinant

Apply the definition of the method of finding determinant of second order

matrix $\left|\begin{array}{cc}2& -7\\ -5& 3\end{array}\right|$.

$\left|\begin{array}{cc}2& -7\\ -5& 3\end{array}\right|=2\times \left(3\right)-\left(-5\right)\left(-7\right)$

- Simplify

To further simplify multiply the values first and then find the difference.

$\begin{array}{c}\left|\begin{array}{cc}2& -7\\ -5& 3\end{array}\right|=2\times \left(3\right)-\left(-5\right)\left(-7\right)\\ =6-35\\ =-29\end{array}$