Question

Evaluate each determinant using expansion by minors.

29.$\left|\begin{array}{ccc}-2& 7& -2\\ 4& 5& 2\\ 1& 0& -1\end{array}\right|$

Short answer

The value of the determinant $\left|\begin{array}{ccc}-2& 7& -2\\ 4& 5& 2\\ 1& 0& -1\end{array}\right|=\text{62}$.

Step-by-step solution

- Decide row to use for expansion

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

$\left|\begin{array}{ccc}-2& 7& -2\\ 4& 5& 2\\ 1& 0& -1\end{array}\right|=1\left|\begin{array}{cc}7& -2\\ 5& 2\end{array}\right|-0\left|\begin{array}{cc}-2& -2\\ 4& 2\end{array}\right|+\left(-1\right)\left|\begin{array}{cc}-2& 7\\ 4& 5\end{array}\right|$

- 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.

$\begin{array}{c}\left|\begin{array}{ccc}-2& 7& -2\\ 4& 5& 2\\ 1& 0& -1\end{array}\right|=1\left|\begin{array}{cc}7& -2\\ 5& 2\end{array}\right|-0\left|\begin{array}{cc}-2& -2\\ 4& 2\end{array}\right|+\left(-1\right)\left|\begin{array}{cc}-2& 7\\ 4& 5\end{array}\right|\\ =\left(14-\left(-10\right)\right)-0\left(-4-\left(-8\right)\right)-\left(-10-28\right)\end{array}$

- Multiply and simplify

Further simplify the values.

$\begin{array}{c}\left|\begin{array}{ccc}-2& 7& -2\\ 4& 5& 2\\ 1& 0& -1\end{array}\right|=\left(14-\left(-10\right)\right)-0\left(-4-\left(-8\right)\right)-\left(-10-28\right)\\ =24-0+38\\ =\text{62}\end{array}$