Question

In Problems 43–50, use properties of determinants to find the value of each determinant if it is known that

$$\begin{gathered} \left|\begin{array}{llr}x& y& z\\ u& v& w\\ 1& 2& 3\end{array}\right|=4 \\ \left|\begin{array}{ccc}x+3& y+6& z+9\\ 3u-1& 3v-2& 3w-3\\ 1& 2& 3\end{array}\right| \end{gathered}$$

Short answer

The value of the determinant is $12$.

Step-by-step solution

Step 1. Given Information 

We are given,

$$\begin{gathered} \left|\begin{array}{llr}x& y& z\\ u& v& w\\ 1& 2& 3\end{array}\right|=4 \\ \left|\begin{array}{ccc}x+3& y+6& z+9\\ 3u-1& 3v-2& 3w-3\\ 1& 2& 3\end{array}\right| \end{gathered}$$

We have to solve the given determinant using properties of determinants.

Step 2. Solving the determinant 

Using subtraction of determinant, we get

$\left|\begin{array}{ccc}x+3& y+6& z+9\\ 3u-1& 3v-2& 3w-3\\ 1& 2& 3\end{array}\right|⟹\left|\begin{array}{ccc}x+3& y+6& z+9\\ 3u& 3v& 3w\\ 1& 2& 3\end{array}\right|-\left|\begin{array}{ccc}x+3& y+6& z+9\\ 1& 2& 3\\ 1& 2& 3\end{array}\right|$

Since the two rows of second determinant is same, so

role="math" $\left|\begin{array}{ccc}x+3& y+6& z+9\\ 3u-1& 3v-2& 3w-3\\ 1& 2& 3\end{array}\right|⟹\left|\begin{array}{ccc}x+3& y+6& z+9\\ 3u& 3v& 3w\\ 1& 2& 3\end{array}\right|-0$

Using the addition of determinant, we get

$=\left|\begin{array}{ccc}x& y& z\\ 3u& 3v& 3w\\ 1& 2& 3\end{array}\right|+\left|\begin{array}{ccc}3& 6& 9\\ 3u& 3v& 3w\\ 1& 2& 3\end{array}\right|$

Step 3. Solving the determinant 

Factoring out the GCF $3$from the determinant, we get

role="math" localid="1647424003126" $$\begin{gathered} =3\left|\begin{array}{ccc}x& y& z\\ u& v& w\\ 1& 2& 3\end{array}\right|+3\left|\begin{array}{ccc}1& 2& 3\\ 3u& 3v& 3w\\ 1& 2& 3\end{array}\right| \\ =3\left|\begin{array}{ccc}x& y& z\\ u& v& w\\ 1& 2& 3\end{array}\right|+3\times 0 \\ =3\times 4 \\ =12 \end{gathered}$$

Hence, the value of the determinant is $12$.