In Exercises 2, determine which matrices are in reduced

echelon form and which others are only in echelon form.

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

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

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

d. \(\left[ {\begin{array}{*{20}{c}}0&1&1&1&1\\0&0&2&2&2\\0&0&0&0&3\\0&0&0&0&0\end{array}} \right]\)

Check whether the provided matrix is in the reduced echelon form or just the echelon form for the given augmented matrices.

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

• The nonzero rows should be positioned above the zero rows.
• Each row's leading entry should be in the column to the right of the row above its leading item.
• In each column, all items below the leading entry should be zero.

For the reduced echelon form, the matrix must follow the above conditions as well as some additional conditions as shown below:

• Each column's components below the leading entry must be zero.
• Each column's leading 1 must be the sole nonzero item.

a.

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

The following can be observed from the above matrix:

• The nonzero rows are positioned above the zero rows.
• Each row's leading entry is in the column to the right of the row above its leading item.
• In each column, all items below the leading entry are zero.
• Each column's components below the leading entry are zero.
• Each column's leading 1 is the sole nonzero item.

Thus, matrix (a) is in the reduced echelon form.

b.

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

The following can be observed from the above matrix:

• The nonzero rows are positioned above the zero rows.
• Each row's leading entry is in the column to the right of the row above its leading item.
• In each column, all items below the leading entry are zero.
• Each column's components below the leading entry are not zero.
• Each column's leading 1 is not the sole nonzero item.

Thus, matrix (b) is in the echelon form.

c.

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

The following can be observed from the above matrix:

Thus, matrix (c) is in the reduced echelon form.

d.

Consider the matrix \(\left[ {\begin{array}{*{20}{c}}0&1&1&1&1\\0&0&2&2&2\\0&0&0&0&3\\0&0&0&0&0\end{array}} \right]\).

The following can be observed from the above matrix:

• The nonzero rows are positioned above the zero rows.
• Each row's leading entry is in the column to the right of the row above its leading item.
• In each column, all items below the leading entry are zero.
• Each column's components below the leading entry are not zero.
• Each column's leading 1 is not the sole nonzero item.

Thus, matrix (d) is in the echelon form.

Hence, matrix (a) is in the reduced echelon form, matrix (c) is not in the echelon form, and matrices (b) and (d) are in the echelon form.