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Elementary matrix operations

In mathematics, an elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation (or column operation). Left multiplication (pre-multiplication) by an elementary matrix represents elementary row operations, while right multiplication (post-multiplication) represents elementary column operations.

Elementary matrix operations play an important role in many matrix algebra applications, such as finding the inverse of a matrix, in Gaussian elimination to reduce a matrix to row echelon form and solving simultaneous linear equations.

    Elementary operations:

  • Interchange two rows (or columns);
  • Multiply each element in a row (or column) by a non-zero number.;
  • Multiply a row (or column) by a non-zero number and add the result to another row (or column).

    Elementary operations notation:

    Compact notation to describe elementary operations:

    Operation descriptionNotation
    Row
    operations
    1. Interchange rows i and jRi ↔ Rj
    2. Multiply row i by s, where s ≠ 0sRi → Ri
    3. Add s times row i to row jsRi + Rj → Rj
    Column
    operations
    1. Interchange columns i and jCi ↔ Cj
    2. Multiply column i by s, where s ≠ 0sCi → Ci
    3. Add s times columns i to columns jsCi + Cj → Cj

Elementary operators

Each type of elementary operation may be performed by matrix multiplication, using square matrices called elementary operators.

Elementary row operations

To perform an elementary row operation on a A, an n×m matrix, take the following steps:
  • To find E, the elementary row operator, apply the operation to an n×n identity matrix.
  • To carry out the elementary row operation, premultiply A by E.
Illustrate this process for each of the three types of elementary row operations.

Interchange two rows

Suppose we want to interchange the first and second rows of A, a 3×2 matrix. To create the elementary row operator E, we interchange the first and second rows of the identity matrix I3:

100 010 001 010 100 001
I3E

Then, to interchange the first and second rows of A, we premultiply A by E (R1 ↔ R2):

010 100 001 42 67 24 = 0·4 + 1·6 + 0·20·2 + 1·7 + 0·4 1·4 + 0·6 + 0·21·2 + 0·7 + 0·4 0·4 + 0·6 + 1·20·2 + 0·7 + 1·4 = 67 42 24
EAB

Multiply a row by a number

Suppose we want to multiply each element in the third row of Matrix A by 3. Assume A is a 3×2 matrix. To create the elementary row operator E, we multiply each element in the third row of the identity matrix I3 by 3:

100 010 001 100 010 003
I3E

Then, to multiply each element in the third row of A by 3, we premultiply A by E (3 R3 → R3):

100 010 003 42 67 24 = 1·4 + 0·6 + 0·21·2 + 0·7 + 0·4 0·4 + 1·6 + 0·20·2 + 1·7 + 0·4 0·4 + 0·6 + 3·20·2 + 0·7 + 3·4 = 42 67 612
EAB

N.B. Divide a row by a number

If we want to divide each element in some row of matrix by number n we must multiply each element of this row by 1n.

Multiply a row and add it to another row

Assume A is a 3×2 matrix. Suppose we want to multiply each element in the first row of A by 4; and we want to add that result to the second row of A. For this operation, creating the elementary row operator is a two-step process. First, we multiply each element in the first row of the identity matrix I2 by 4. Next, we add the result of that multiplication to the second row of I2 to produce E:

10 01 10 0 + 4·11 + 4·0 = 10 41
I2E

Then, to multiply each element in the first row of A by 4 and add that result to the second row, we premultiply A by E ( 4 R1 + R2 → R2):

10 41 42 67 = 1·4 + 0·61·2 + 0·7 4·4 + 1·64·2 + 1·7 = 42 2215
EAB

Elementary column operations

To perform an elementary column operation on a A, an n×m matrix, take the following steps:
  • To find E, the elementary column operator, apply the operation to an m×m identity matrix.
  • To carry out the elementary column operation, postmultiply A by E.

Examples of elementary matrix operations

Example 1.
Use elementary row operations to convert matrix A to the upper triangular matrix
A =  (  4  2  0  )
 1  3  2 
 -1  3  10 

Solution:

R1 ↔ R2 (interchange the first and second rows)

(  4  2  0  )  ~  (  1  3  2  )  ~ 
 1  3  2  4  2  0 
 -1  3  10  -1  3  10 

-4 R1 + R2 → R2 (multiply 1 row by -4 and add it to 2 row); R1 + R3 → R3 (1 row add to 3 row)

 ~  (  1  3  2  )  ~  (  1  3  2  )  ~ 
 4 + (-4)·1  2 + (-4)·3  0 + (-4)·2  0  -10  -8 
 -1 + 1  3 + 3  10 + 2  0  6  12 

R2 / (-2) → R2 (divide 2 row by -2); R3 / 6 → R3 (divide 3 row by 6)

 ~  (  1  3  2  )  ~  (  1  3  2  )  ~ 
 0  -10/(-2)  -8/(-2)  0  5  4 
 0  6/6  12/6  0  1  2 

R2 ↔ R3 (interchange the 2 row and 3 row)

 ~  (  1  3  2  )  ~ 
 0  1  2 
 0  5  4 

-5 R2 + R3 → R3 (multiply 2 row by -5 and add it to 3 row);

 ~  (  1  3  2  )  ~  (  1  3  2  )
 0  1  2  0  1  2 
 0  5 + (-5)·1  4 + (-5)·2  0  0  -6 

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