Question

In the following exercise, evaluate each determinant by expanding by minors along the first row.

$\left|\begin{array}{rrr}-2& -3& -4\\ 5& -6& 7\\ -1& 2& 0\end{array}\right|$

Short answer

The value of the given determinant is$33$.

Step-by-step solution

Step 1. Given information

The given determinant is :

$\left|\begin{array}{rrr}-2& -3& -4\\ 5& -6& 7\\ -1& 2& 0\end{array}\right|$

Step 2. Expanding the determinant along the first row by minors

The expansion of general determinant along the first row is given as:$\left|\begin{array}{lll}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|=a_1\left|\begin{array}{ll}{b}_2& c_2\\ b_3& c_3\end{array}\right|-b_1\left|\begin{array}{ll}{a}_2& c_2\\ a_3& c_3\end{array}\right|+c_1\left|\begin{array}{ll}{a}_2& b_2\\ a_3& b_3\end{array}\right|$

Similarly, for the given determinant, we have:

localid="1644588932923" $\left|\begin{array}{ccc}-2& -3& -4\\ 5& -6& 7\\ -1& 2& 0\end{array}\right|=-2\left|\begin{array}{cc}-6& 7\\ 2& 0\end{array}\right|+3\left|\begin{array}{cc}5& 7\\ -1& 0\end{array}\right|-4\left|\begin{array}{cc}5& -6\\ -1& 2\end{array}\right|$

Step 3. evaluating the determinant

Evaluating the determinant, we get:

$$\begin{gathered} =-2\left|\begin{array}{cc}-6& 7\\ 2& 0\end{array}\right|+3\left|\begin{array}{cc}5& 7\\ -1& 0\end{array}\right|-4\left|\begin{array}{cc}5& -6\\ -1& 2\end{array}\right| \\ =-2[-6\times 0-7\times 2]+3[5\times 0-(-1\times 7\left)\right]-4[5\times 2-(-6\times -1\left)\right] \\ =-2[0-14]+3\left[7\right]-4[10-6] \\ =28+21-40 \\ =33 \end{gathered}$$