Write the reduced echelon form of a \(3 \times 3\) matrix A such that the first two columns of Aare pivot columns and

\(A = \left( {\begin{aligned}{*{20}{c}}3\\{ - 2}\\1\end{aligned}} \right) = \left( {\begin{aligned}{*{20}{c}}0\\0\\0\end{aligned}} \right)\).

Suppose the first two columns of Aare pivot columns.

\(E = \left( {\begin{aligned}{*{20}{c}}1&0& * \\0&1& * \\0&0&0\end{aligned}} \right)\)

Step 2: Determine the reduced echelon form of the matrix by inspection

The reduced echelon form of matrix \(A\) appears as \(E = \left( {\begin{aligned}{*{20}{c}}1&0& * \\0&1& * \\0&0&0\end{aligned}} \right)\). The solution to the equation \(E{\mathop{\rm x}\nolimits} = 0\) is the same as that of \(Ax = 0\) since \(E\) is row equivalent to \(A\).

\(\left( {\begin{aligned}{*{20}{c}}1&0& * \\0&1& * \\0&0&0\end{aligned}} \right)\left( {\begin{aligned}{*{20}{c}}3\\{ - 2}\\1\end{aligned}} \right) = \left( {\begin{aligned}{*{20}{c}}0\\0\\0\end{aligned}} \right)\)

The reduced echelon form of a matrix by inspection is \(E = \left( {\begin{aligned}{*{20}{c}}1&0&{ - 3}\\0&1&2\\0&0&0\end{aligned}} \right)\).

Thus, the reduced echelon form of the matrix is \(E = \left( {\begin{aligned}{*{20}{c}}1&0&{ - 3}\\0&1&2\\0&0&0\end{aligned}} \right)\).