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Elementary Matrices
Def. Let 𝐴 be an 𝑚 × 𝑛 matrix. Any of the following are called elementary row
(column) operations.
1. Interchanging two rows (columns) of 𝐴.
2. Multiply a row (column) by a nonzero real number.
3. Replace a row (column) by its sum with a multiple of another row (column).
1 −2 3 5
Ex. Let 𝐴 = [2 −3 4 1]
6 0 1 2
a. An example of an elementary row operation of type 1 on 𝐴:
interchange rows 1 and 3 which we will denote 𝑅1 ↔ 𝑅3.
6 0 1 2
𝐵 = [2 −3 4 1]
1 −2 3 5
b. An example of an elementary row operation of type 2 on 𝐴:
multiply the second row of 𝐴 by −2, denoted −2𝑅2 → 𝑅2.
1 −2 3 5
𝐶 = [−4 6 −8 −2 ]
6 0 1 2
c. An example of an elementary row operation of type 3 on 𝐴:
replace row one with row one minus 2 times row three, denoted
𝑅1 − 2𝑅3 → 𝑅1.
−11 −2 1 1
𝐷=[ 2 −3 4 1].
6 0 1 2
Elementary Matrices
Def. Let 𝐴 be an 𝑚 × 𝑛 matrix. Any of the following are called elementary row
(column) operations.
1. Interchanging two rows (columns) of 𝐴.
2. Multiply a row (column) by a nonzero real number.
3. Replace a row (column) by its sum with a multiple of another row (column).
1 −2 3 5
Ex. Let 𝐴 = [2 −3 4 1]
6 0 1 2
a. An example of an elementary row operation of type 1 on 𝐴:
interchange rows 1 and 3 which we will denote 𝑅1 ↔ 𝑅3.
6 0 1 2
𝐵 = [2 −3 4 1]
1 −2 3 5
b. An example of an elementary row operation of type 2 on 𝐴:
multiply the second row of 𝐴 by −2, denoted −2𝑅2 → 𝑅2.
1 −2 3 5
𝐶 = [−4 6 −8 −2 ]
6 0 1 2
c. An example of an elementary row operation of type 3 on 𝐴:
replace row one with row one minus 2 times row three, denoted
𝑅1 − 2𝑅3 → 𝑅1.
−11 −2 1 1
𝐷=[ 2 −3 4 1].
6 0 1 2
