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

The matrix is in 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 \(A = \left[ {\begin{array}{*{20}{c}}1&0&1\\0&1&0\\1&1&1\end{array}} \right]\).

Here, the non-zero row is not positioned above zero rows.

Consider \(b = \left[ {\begin{array}{*{20}{c}}0\\0\\1\end{array}} \right]\) is the required vector.

In \(\left( {A|b} \right)\) form, it can be written as shown below:

\(\left[ {\begin{array}{*{20}{c}}1&0&1&0\\0&1&0&0\\1&1&1&1\end{array}} \right]\)

Apply the row operation in the matrix \(\left[ {\begin{array}{*{20}{c}}1&0&1&0\\0&1&0&0\\1&1&1&1\end{array}} \right]\).

Use rows one and two to eliminate \({x_1}\), \({x_2}\), and \({x_3}\) from the third equation.

\(\left[ {\begin{array}{*{20}{c}}1&0&1&0\\0&1&0&0\\0&0&0&1\end{array}} \right]\)

It shows that column \(b = \left[ {\begin{array}{*{20}{c}}0\\0\\1\end{array}} \right]\) of the assumed matrix does not span \({\mathbb{R}^3}\).

Therefore, the required matrix is \(A = \left[ {\begin{array}{*{20}{c}}1&0&1\\0&1&0\\1&1&1\end{array}} \right]\).