An Echelon Form of a Matrix
Definition. An [katex]m \times n[/katex] matrix A is said to be in (row) echelon form (or an echelon matrix) if it has the following properties: (i) All nonzero rows are above any zero rows (consisting of all zeros). (ii) The first nonzero entry in cach nonzero row is to the right of the first nonzero entry of each preceding row. That is, the number of zeros occurring before the first nonzero entry in each nonzero row is greater than the number of zeros that appear before the first nonzero element in any preceding row.
Earn Money online without investment Earn Money online from home without investment In an echelon matrix, the first nonzero entry of a row is called a pivot (or a row leader). There is at the most one pivot in each row and in cach column of an echelon matrix. A column containing a pivot is called a pivot column.
An echelon matrix in which each pivot is 1 and every other entry of the pivot column is zero, is said to be in row reduced echelon form. The matrices
\left[\begin{array}{ll}
0 & 5&2&1\\
0&0&9&-1\\
0&0&0&0
\end{array}\right]and
\left[\begin{array}{ll}
1&3 &0&0\\
0&0&1&0\\
0&0&0&1\\
0&0&0&0\\
\end{array}\right]are in echelon form. The second matrix is in row reduced echelon form.
Every matrix is row equivalent to a matrix in echelon form (reduced echelon form).
Theorem. Every matrix is row equivalent to a matrix in echelon form (reduced echelon form).
Proof. Let A be a given [katex]m \times n[/katex] matrix. Either every clement in the first column of A is zero or there exists a nonzero element a (say) in the kth row, of this column. In the second case, interchanging the first and k th rows of A, we obtain a matrix B whose first clement [katex]b_{11}=a \neq 0[/katex]. Multiplying the elements of the flrst fow of B by [katex]a^{-1}[/katex], we obtain a matrix C whose first entry [katex]c_{11}=1[/katex]. By adding proper multiples of the first row of C to other rows, the remaining elements of the first column can be made zeros. In the first case, we consider the second column and so on till we find a column with a nonzero entry. We then repeat the process as for the second case. Thus A is row equivalent to a matrix in either of the forms
\left[\begin{array}{ll}
0 & C
\end{array}\right] \text { or }\left[\begin{array}{ll}
1 & D \\
0 & E
\end{array}\right] \text {. }where, in the first case, [katex]\theta[/katex] represents the [katex]m \times 1[/katex] zero matrix and C an [katex]m \times(n-1)[/katex] matrix.
In the second case [katex]\theta[/katex] represents the [katex](m-1) \times 1[/katex] zero matrix, [katex]D[/katex] a [katex]1 \times(n-1)[/katex] matrix and E an [katex](m-1) \times(n-1)[/katex] matrix. In the first case repeat the above process with the matrix C, whereas in the second case repeat the process with the matrix E. A continuation of this process leads to the desired form.
When a matrix is in echclon form, then by adding suitable multiplies of second, third, [katex]\cdots[/katex] rows to the first, second, ‘. rows, in succession, we get the reduced echelon form.
Example of An Echelon Form of a Matrix
Example: Reduce the matrix
A=\left[\begin{array}{rrr}
6 & 3 & -4 \\
-4 & 1 & -6 \\
1 & 2 & -5
\end{array}\right] \quad \text { into the echelon form. }Solution. Here
A \stackrel{R}{\sim}\left[\begin{array}{ccc}
1 & 2 & -5 \\
-4 & 1 & -6 \\
6 & 3 & -4
\end{array}\right] \quad \text { by } R_{13}\stackrel{R}{\sim}\left[\begin{array}{ccc}1 & 2 & -5 \\ 0 & 9 & -26 \\ 0 & -9 & \,\,\ 26\end{array}\right] \quad \text { by } R_2+4 R_1 \text { and } R_3-6 R_1\stackrel{R}{\sim} \left[\begin{array}{ccc}1 & 2 & -5 \\ 0 & 1 & -\frac{26}{9} \\ 0 & 0 & 0\end{array}\right] \text { by } R_3+R_2\stackrel{R}{\sim}\left[\begin{array}{ccc}1 & 2 & -5 \\ 0 & 1 & -\frac{26}{9} \\ 0 & 0 & 0\end{array}\right] \text { by } \frac{1}{9}R_2which is the desired echelon form.