Question

Evaluate the determinant $$\begin{gathered} \left[-2-1-3 \\ -122 \\ 4-40\right] \end{gathered}$$by expanding by minors.

Short answer

The value of the determinant is -12.

Step-by-step solution

Step 1. Given information

The given determinant is$$\begin{gathered} \left[-2-1-3 \\ -122 \\ 4-40\right] \end{gathered}$$

Step 2. Calculation

Expand the determinant using second row.

Be careful with the signs

$$\begin{gathered} \left[\begin{array}{ccc}+& -& +\\ -& +& -\\ +& -& +\end{array}\right] \\ -(-1)\left[\begin{array}{cc}-1& -3\\ -4& 0\end{array}\right]+2\left[\begin{array}{cc}-2& -3\\ 4& 0\end{array}\right]-2\left[\begin{array}{cc}-2& -1\\ 4& -4\end{array}\right] \\ =1\left[(-1\times 0)-(-3\times -4)\right]+2\left[(-2\times 0)-(-3\times 4)\right]-2\left[(-2\times -4)-(-1\times 4)\right] \\ =1(0-12)+2(+12)-2(8+4) \\ =-12+24-24 \\ =-12 \end{gathered}$$