Question

Evaluate each determinant using expansion by minors.

$\left|\begin{array}{ccc}2& 3& 4\\ 6& 5& 7\\ 1& 2& 8\end{array}\right|$

Short answer

The value of the determinant $\left|\begin{array}{ccc}2& 3& 4\\ 6& 5& 7\\ 1& 2& 8\end{array}\right|=-43$.

Step-by-step solution

- Decide row to use for expansion

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

$\left|\begin{array}{ccc}2& 3& 4\\ 6& 5& 7\\ 1& 2& 8\end{array}\right|=2\left|\begin{array}{cc}5& 7\\ 2& 8\end{array}\right|-3\left|\begin{array}{cc}6& 7\\ 1& 8\end{array}\right|+4\left|\begin{array}{cc}6& 5\\ 1& 2\end{array}\right|$

- 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.

$\begin{array}{c}\left|\begin{array}{ccc}2& 3& 4\\ 6& 5& 7\\ 1& 2& 8\end{array}\right|=2\left|\begin{array}{cc}5& 7\\ 2& 8\end{array}\right|-3\left|\begin{array}{cc}6& 7\\ 1& 8\end{array}\right|+4\left|\begin{array}{cc}6& 5\\ 1& 2\end{array}\right|\\ =2\left(40-14\right)-3\left(48-7\right)+4\left(12-5\right)\end{array}$

- Multiply and simplify

Further simplify the values.

$\begin{array}{c}\left|\begin{array}{ccc}2& 3& 4\\ 6& 5& 7\\ 1& 2& 8\end{array}\right|=2\left(40-14\right)-3\left(48-7\right)+4\left(12-5\right)\\ =2\times 26-3\times 41+4\times 7\\ =52-123+28\\ =-43\end{array}$