Question

Evaluate each determinant using expansion by minors.

31.$\left|\begin{array}{ccc}1& 5& -4\\ -7& 3& 2\\ 6& 3& -1\end{array}\right|$

Short answer

The value of the determinant $\left|\begin{array}{ccc}1& 5& -4\\ -7& 3& 2\\ 6& 3& -1\end{array}\right|=172$.

Step-by-step solution

- Decide row to use for expansion

In this case, use first row for the expansion by minors.

$\left|\begin{array}{ccc}1& 5& -4\\ -7& 3& 2\\ 6& 3& -1\end{array}\right|=\left(1\right)\left|\begin{array}{cc}3& 2\\ 3& -1\end{array}\right|-5\left|\begin{array}{cc}-7& 2\\ 6& -1\end{array}\right|+\left(-4\right)\left|\begin{array}{cc}-7& 3\\ 6& 3\end{array}\right|$

- Evaluate 2×2 determinants

The determinant of second order matrix is found by calculating the difference of the product of the two diagonals, that is., $\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|=ad-bc$. Apply this definition to find the $2\times 2$determinant.

$\begin{array}{c}\left|\begin{array}{ccc}1& 5& -4\\ -7& 3& 2\\ 6& 3& -1\end{array}\right|=\left(1\right)\left|\begin{array}{cc}3& 2\\ 3& -1\end{array}\right|-5\left|\begin{array}{cc}-7& 2\\ 6& -1\end{array}\right|+\left(-4\right)\left|\begin{array}{cc}-7& 3\\ 6& 3\end{array}\right|\\ =\left(-3-6\right)-5\left(7-12\right)-4\left(-21-18\right)\end{array}$

- Multiply and simplify

Further simplify the values.

$\begin{array}{c}\left|\begin{array}{ccc}1& 5& -4\\ -7& 3& 2\\ 6& 3& -1\end{array}\right|=\left(-3-6\right)-5\left(7-12\right)-4\left(-21-18\right)\\ =-9-5\left(-5\right)-4\left(-39\right)\\ =-9+25+156\\ =172\end{array}$