Construct a \(3 \times 3\) matrix, not in echelon form, whose columns span \({\mathbb{R}^3}\). Show that the matrix you construct has the desired property.

The matrix is in the echelon form if it satisfies the following conditions:

• Non-zero rows should be positioned above zero rows.
• Each row's leading entry should bein the column to the right of the row above its leading item.
• In each column, all items below the leading entry must be zero.

Assume that the matrix is \(A = \left[ {\begin{array}{*{20}{c}}0&0&3\\0&4&0\\5&0&0\end{array}} \right]\).

Here, each row's leading entry is not in the column to the right of the row above its leading item. So, the matrix is in non-echelon form.

Suppose \({\bf{b}} = \left[ {\begin{array}{*{20}{c}}{{b_1}}\\{{b_2}}\\{{b_3}}\end{array}} \right]\) is the required vector.

Re-arrange the assumed vector as shown below:

\(\begin{array}{c}{\bf{b}} = \left[ {\begin{array}{*{20}{c}}{{b_1}}\\{{b_2}}\\{{b_3}}\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}{0 + 0 + {b_1}}\\{0 + {b_2} + 0}\\{{b_3} + 0 + 0}\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}0\\0\\{{b_3}}\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}0\\{{b_2}}\\0\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}{{b_1}}\\0\\0\end{array}} \right]\end{array}\)

Simplify vector \({\bf{b}} = \left[ {\begin{array}{*{20}{c}}0\\0\\{{b_3}}\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}0\\{{b_2}}\\0\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}{{b_1}}\\0\\0\end{array}} \right]\) further.

\(\begin{array}{c}{\bf{b}} = \left[ {\begin{array}{*{20}{c}}0\\0\\{\frac{{5{b_3}}}{5}}\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}0\\{\frac{{4{b_2}}}{4}}\\0\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}{\frac{{3{b_1}}}{3}}\\0\\0\end{array}} \right]\\ = \frac{{{b_3}}}{5}\left[ {\begin{array}{*{20}{c}}0\\0\\5\end{array}} \right] + \frac{{{b_2}}}{4}\left[ {\begin{array}{*{20}{c}}0\\4\\0\end{array}} \right] + \frac{{{b_1}}}{3}\left[ {\begin{array}{*{20}{c}}3\\0\\0\end{array}} \right]\end{array}\)

Thus, it shows that the columns of the assumed matrix span are \({\mathbb{R}^3}\).

Therefore, the required matrix is \(A = \left[ {\begin{array}{*{20}{c}}0&0&3\\0&4&0\\5&0&0\end{array}} \right]\).