Evaluate $4 \times 4$ determinant. $$\left| \begin{array} { c c c c }
1 & - 3 & 4 & 3 \
3 & - 1 & 0 & 6 \
- 4 & 6 & 0 & - 3 \
- 1 & 1 & 1 & 8
\end{array} \right|$$ Use the properties of determinants to your advantage.
A) 330
B) 250
C) 466

Question

Evaluate $4 \times 4$ determinant. $$\left| \begin{array} { c c c c } 
1 & - 3 & 4 & 3 \
3 & - 1 & 0 & 6 \
- 4 & 6 & 0 & - 3 \
- 1 & 1 & 1 & 8
\end{array} \right|$$ Use the properties of determinants to your advantage.
A) 330
B) 250
C) 466

Quizplus · Accepted Answer

We can use the properties of determinants to simplify the computation of this $4 \times 4$ determinant.

First, let's perform row operations to get many 0's in the first column, which will make computing the determinant much easier. The determinant isn't changed when we perform row operations, so the value of the determinant will be the same as the value of the determinant of the matrix we get after performing row operations.

Note that adding 3 times the first row to the second row and adding 4 times the first row to the third row produces a matrix with a diagonal of $1$'s and $0$'s in the first column. Subtracting the fourth row from the third row doesn't affect the value of determinant, either.

$$ \left| \begin{array} { c c c c } 
1 & - 3 & 4 & 3 \
0 & 8 & - 12 & 15 \
0 & 15 & - 16 & 6 \
- 1 & 1 & 1 & 8
\end{array} \right| $$

Now we can expand the determinant along the first column:

\begin{align*}
\left| \begin{array} { c c c c } 
1 & - 3 & 4 & 3 \
0 & 8 & - 12 & 15 \
0 & 15 & - 16 & 6 \
- 1 & 1 & 1 & 8
\end{array} \right| &= 1 \cdot \left| \begin{array} { c c c } 
8 & - 12 & 15 \
15 & - 16 & 6 \
1 & 1 & 8
\end{array} \right| - 3 \cdot \left| \begin{array} { c c c } 
0 & - 12 & 15 \
0 & - 16 & 6 \
-1 & 1 & 8
\end{array} \right| + 4 \cdot \left| \begin{array} { c c c } 
0 & 8 & - 12 \
0 & 15 & - 16 \
-1 & 1 & 1
\end{array} \right| + 3 \cdot \left| \begin{array} { c c c } 
0 & 8 & - 12 \
0 & 15 & - 16 \
-1 & 1 & 1
\end{array} \right| \
&= 1 \cdot (- 931) - 3 \cdot (- 204) + 4 \cdot 448 + 3 \cdot 153 \
&= \boxed{466}
\end{align*}

Evaluate 4×44 \times 44×4 Determinant ∣1−3433−106−460−3−1118∣\left| \begin{array} { c c c c } 1 & - 3 & 4 & 3 \\3 & - 1 & 0 & 6 \\- 4 & 6 & 0 & - 3 \\- 1 & 1 & 1 & 8\end{array} \right|​13−4−1​−3−161​4001​36−38​​

Evaluate $4 \times 4$ Determinant $\left| \begin{array} { c c c c } 1 & - 3 & 4 & 3 \\3 & - 1 & 0 & 6 \\- 4 & 6 & 0 & - 3 \\- 1 & 1 & 1 & 8\end{array} \right|$