Question

Evaluate each determinant by expanding by minors.

$\left|\begin{array}{ccc}2& -4& -3\\ 5& -1& -4\\ 3& 2& 0\end{array}\right|$

Short answer

The required value of the determinant is$25$

Step-by-step solution

Step 1. Given information

The determinant is$\left|\begin{array}{ccc}2& -4& -3\\ 5& -1& -4\\ 3& 2& 0\end{array}\right|$

Step 2. Expand by minors along the third row to evaluate determinant

$$\begin{gathered} \left|\begin{array}{ccc}2& -4& -3\\ 5& -1& -4\\ 3& 2& 0\end{array}\right| \\ =3\left|\begin{array}{cc}-4& -3\\ -1& -4\end{array}\right|-2\left|\begin{array}{cc}2& -3\\ 5& -4\end{array}\right|+0\left|\begin{array}{cc}2& -4\\ 5& -1\end{array}\right| \\ =3\left[\right(-4\left)\right(-4)-(-3\left)\right(-1\left)\right]-2\left[\right(2\left)\right(-4)-(-3\left)\right(5\left)\right]+0 \\ =3\left[13\right]-2\left[7\right]+0 \\ =39-14+0 \\ =25 \end{gathered}$$

Step 3. Required answer

The required value of the determinant is$25$