Echelon matrix

2. Echelon Matrices
• A row of any matrix is
called zero if all its entries
are zero. Now a matrix is
in echelon form if
• (i) All zero rows come
below the non‐zero rows
• (ii) The first nonzero
entry in each nonzero row
is 1 and occurs in a
column to the right of the
leading 1 in the row above
it.
𝐴 =
1 1 0
0 1 1
0 0 1
𝐵 =
1 3 −1
0 1 3
0 0 0
𝐶 =
1 3 0
0 0 1
0 0 0
𝐷 =
1 0 3 4
0 0 1 4
0 0 0 0
are echelon matrices.

3. Reduced
Echelon
Matrix
1 3 0 0
0 0 1 0
0 0 0 1
1 0 0
0 1 2
and
1 0 0 0
0 1 0 0
0 0 0 0
.

4. EXAMPLE
Given A =
1 4 −2
2 7 −1
2 9 −7
R2 − 2R1, R3 − 2R1
𝐴 ≈
1 4 −2
0 −1 3
0 1 −3
𝑅3 + 𝑅2
≈
1 4 −2
0 −1 3
0 0 0
−1 𝑅2
≈
1 4 −2
0 1 −3
0 0 0
which is in echelon/reduced echelon form.

5. EXAMPLE
1 2 1
3 1 2
0 1 2
≈ 𝐼3 ⋅
𝐴 =
1 2 1
3 1 2
0 1 2
𝑅2 + −3 𝑅1
≈
1 2 1
0 −5 −1
0 1 2
R23
≈
1 2 1
0 1 2
0 −5 −1
R1 + −2 𝑅2, 𝑅3 +
5𝑅2 ≈
1 0 −3
0 1 2
0 0 9
1
9
𝑅3
≈
1 0 −3
0 1 2
0 0 1
𝑅1 + 3𝑅3, 𝑅2 +
−2 𝑅3 ≈
1 0 0
0 1 0
0 0 1
= 𝐼3 ⋅
Hence the given matrix has been reduced
to the identity matrix.

6. THE END