Question

find the value of each determinant

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

Short answer

$D=108$

Step-by-step solution

Step 1. The given information is

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

Step 2. Using the formula for the determinant D=abcdefghiD=a(ei-fh)-b(di-gf)-c(dh-eg)

$$\begin{gathered} D=\left|\begin{array}{ccc}1& 4& 0\\ -1& 2& 6\\ 4& 1& 3\end{array}\right|\begin{array}{}\\ \\ \end{array} \\ D=1(2\times 3-6\times 1)-4(-1\times 3-4\times 6)-0(-1\times 1-2\times 4) \\ D=1(6-6)-4(-3-24)-0(-1-8) \\ D=1\left(0\right)-4(-27)-0\left(9\right) \\ D=108 \end{gathered}$$