Q. 13 - Precalculus Enhanced with Graphing Utilities

Question

Find the value of the determinant.

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

Step-by-Step Solution

Answer

$\left|\begin{array}{ccc}4& -1& 2\\ 6& -1& 0\\ 1& -3& 4\end{array}\right|=-26$

Step 1. Given information

We have been given:

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

We have to find the value of this determinant. 

Step 2. Use formula a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 = a 11 a 22 a 23 a 32 a 33 - a 12 a 21 a 23 a 31 a 33 + a 13 a 21 a 22 a 31 a 32 .

$\left|\begin{array}{ccc}4& -1& 2\\ 6& -1& 0\\ 1& -3& 4\end{array}\right|=4\cdot \left|\begin{array}{l}-1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\\ -3\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}4\end{array}\right|-(-1)\cdot \left|\begin{array}{cc}6& 0\\ 1& 4\end{array}\right|+2\cdot \left|\begin{array}{cc}6& -1\\ 1& -3\end{array}\right|$

Step 3. Simplify

$4\cdot \left|\begin{array}{l}-1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\\ -3\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}4\end{array}\right|-(-1)\cdot \left|\begin{array}{cc}6& 0\\ 1& 4\end{array}\right|+2\cdot \left|\begin{array}{cc}6& -1\\ 1& -3\end{array}\right|$

$$\begin{gathered} =4\cdot \left[\right(-1)\cdot 4-(-3)\cdot 0]+1\cdot [6\cdot 4-1\cdot 0]+2\cdot [6\cdot (-3)-1\cdot (-1\left)\right] \\ =4\cdot (-4)+1\cdot 24+2\cdot (-18+1) \\ =-16+24+2\cdot (-17) \\ =8-34 \\ =-26 \end{gathered}$$