91Ó°ÊÓ

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.

Consider the condition that the matrix has nonzero rows, and at the leftmost column of the matrix, the nonzero entry is the leading entry at the top. Here, hasa nonzero value and \( * \) has any value including 0.

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

Consider the condition that nonzero rows should be stacked on top of rows with all zeros,and there should be any nonzero value in the leading entries in the column.

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

Consider the condition that nonzero rows should be stacked on top of rows with all zerosand they should contain a single leading entry.

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

Thus, the possible echelon forms of a nonzero \(2 \times 2\) matrix are \(\left[ {\begin{array}{*{20}{c}} \square & * \\ 0&\square \end{array}} \right]\), \(\left[ {\begin{array}{*{20}{c}} \square & * \\ 0&0 \end{array}} \right]\), and \(\left[ {\begin{array}{*{20}{c}} 0&\square \\ 0&0 \end{array}} \right]\)

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