Let $A=\begin{bmatrix}1&0&0\\2&1&0\\3&2&1\end{bmatrix}$ and $U_1,U_2,U_3$ be column matrices satisfying $AU_1=\begin{bmatrix}1\\0\\0\end{bmatrix},AU_2=\begin{bmatrix}2\\3\\6\end{bmatrix},AU_3=\begin{bmatrix}2\\3\\1\end{bmatrix}$. If $U$ is 3 × 3 matrix whose columns are $U_1,U_2,U_3$, then $|U|=$
Answer & explanation
Correct answer: option 1
$A = \begin{bmatrix}1&0&0\\2&1&0\\3&2&1\end{bmatrix}$
$AU_1 = \begin{bmatrix}1\\0\\0\end{bmatrix}, \quad AU_2 = \begin{bmatrix}2\\3\\6\end{bmatrix}, \quad AU_3 = \begin{bmatrix}2\\3\\1\end{bmatrix}$
$A[U_1 \; U_2 \; U_3] = [AU_1 \; AU_2 \; AU_3]$
$A U = B$
$|A||U| = |B|$
$|U| = \frac{|B|}{|A|}$
$|A| = 1$
$B = \begin{bmatrix}1&2&2\\0&3&3\\0&6&1\end{bmatrix}$
$|B| = 1\begin{vmatrix}3&3\\6&1\end{vmatrix}$
$= 3\cdot 1 - 3\cdot 6$
$= 3 - 18 = -15$
$|U| = \frac{-15}{1} = -15$
$\textbf{The value of } |U| \textbf{ is } -15.$