Question

Evaluate the determinant.
$\left|\begin{array}{ccc}3& -4& 5\\ 1& 1& -2\\ 2& 3& 1\end{array}\right|$

Answer 1

Let $A=\left|\begin{array}{ccc}3& -4& 5\\ 1& 1& -2\\ 2& 3& 1\end{array}\right|$
By expanding along $R_1$ (first row), we get
$|A|=3\left|\begin{array}{cc}1& -2\\ 3& 1\end{array}\right|-(-4)\left|\begin{array}{cc}1& -2\\ 2& 1\end{array}\right|+5\left|\begin{array}{cc}1& 1\\ 2& 3\end{array}\right|=3(1+6)+4(1+4)+5(3-2)=3\left(7\right)+4\left(5\right)+5\left(1\right)=21+20+5=46$