30. The value of $| \mathrm { U } |$ is\\
(A) 3\\
(B) - 3\\
(C) $3 / 2$\\
(D) 2
Sol. (A)\\
Let $U _ { 1 }$ be $\left[ \begin{array} { c } x \\ y \\ z \end{array} \right]$ so that\\
$\left[ \begin{array} { l l l } 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{array} \right] \left[ \begin{array} { l } x \\ y \\ z \end{array} \right] = \left[ \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right] \Rightarrow \left[ \begin{array} { l } x \\ y \\ z \end{array} \right] = \left[ \begin{array} { c } 1 \\ - 2 \\ 1 \end{array} \right]$\\
Similarly $U _ { 2 } = \left[ \begin{array} { c } 2 \\ - 1 \\ - 4 \end{array} \right] , U _ { 3 } = \left[ \begin{array} { c } 2 \\ - 1 \\ - 3 \end{array} \right]$.\\
Hence $U = \left[ \begin{array} { c c c } 1 & 2 & 2 \\ - 2 & - 1 & - 1 \\ 1 & - 4 & - 3 \end{array} \right]$ and $| U | = 3$.\\