Question

Find the value of the determinant.

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

Short answer

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

Step-by-step solution

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 a11a12a13a21a22a23a31a32a33=a11a22a23a32a33-a12a21a23a31a33+a13a21a22a31a32.

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

Step 3. Simplify

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

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