Matrices may be long.

Algebra Level 4

If

\(A = \begin{bmatrix} 1 & 0 & 0 \\
2 & 1 & 0 \\ 3 & 2 & 1 \\ \end{bmatrix}\)

and \(U_1 , U_2 \) and \(U_3\) are column matrices satisfying

\(AU_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \\ \end{bmatrix} \)

\(AU_2 = \begin{bmatrix} 2 \\ 3 \\ 0 \\ \end{bmatrix} \)

\(AU_3 = \begin{bmatrix} 2 \\ 3 \\ 1 \\ \end{bmatrix} \)

and \(U\) is a \(3 \times 3\) matrix whose columns \(C_1 , C_2 , C_3\) are \(U_1 , U_2 , U_3\) respectively, then the value of \(|U|\) is

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