Question

Evaluate each determinant using expansion by minors.

$\left|\begin{array}{ccc}7& 3& -4\\ -2& 9& 6\\ 0& 0& 0\end{array}\right|$

Short answer

The value of the determinant $\left|\begin{array}{ccc}7& 3& -4\\ -2& 9& 6\\ 0& 0& 0\end{array}\right|=0$.

Step-by-step solution

- Decide row to use for expansion

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

$\left|\begin{array}{ccc}7& 3& -4\\ -2& 9& 6\\ 0& 0& 0\end{array}\right|=0\left|\begin{array}{cc}3& -4\\ 9& 6\end{array}\right|-0\left|\begin{array}{cc}7& -4\\ -2& 6\end{array}\right|+0\left|\begin{array}{cc}7& 3\\ -2& 9\end{array}\right|$

- 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.

localid="1648216109992" $\begin{array}{c}\left|\begin{array}{ccc}7& 3& -4\\ -2& 9& 6\\ 0& 0& 0\end{array}\right|=0\left|\begin{array}{cc}3& -4\\ 9& 6\end{array}\right|-0\left|\begin{array}{cc}7& -4\\ -2& 6\end{array}\right|+0\left|\begin{array}{cc}7& 3\\ -2& 9\end{array}\right|\\ =0\left(18-\left(-36\right)\right)-0\left(42-8\right)+0\left(63-\left(-6\right)\right)\end{array}$

- Multiply and simplify

Further simplify the values.

$\begin{array}{c}\left|\begin{array}{ccc}7& 3& -4\\ -2& 9& 6\\ 0& 0& 0\end{array}\right|=0\left(18-\left(-36\right)\right)-0\left(42+8\right)+0\left(63-\left(-6\right)\right)\\ =0\end{array}$