3.8 Elementary Matrices

MATH0005 Algebra 1

1. About these notes
2. 1 Logic
3. 2 Sets and functions
4. 3 Matrices
   1. 3.1 Matrix definitions
   2. 3.2 Matrix multiplication
   3. 3.3 Transpose
   4. 3.4 Multiplication properties
   5. 3.5 Invertible matrices
   6. 3.6 Systems of linear equations
   7. 3.7 Row operations
   8. 3.8 Elementary matrices
      1. 3.8.1 Definition of an elementary matrix
      2. 3.8.2 Doing a row operation is the same as multiplying by an elementary matrix
   9. 3.9 Row reduced echelon form
   10. 3.10 RREF existence and uniqueness
   11. 3.11 Solving RREF systems
   12. 3.12 Invertibility and RREF
   13. 3.13 Finding inverses
   14. Further reading
5. 4 Linear algebra

3.8 Elementary matrices

3.8.1 Definition of an elementary matrix

An elementary matrix is one you can get by doing a single row operation to an identity matrix.

Example 3.8.1.

• •

  The elementary matrix (0110) results from doing the row operation 𝐫1↔𝐫2 to I2.

• •

  The elementary matrix (120010001) results from doing the row operation 𝐫1↦𝐫1+2⁢𝐫2 to I3.

• •

  The elementary matrix (−1001) results from doing the row operation 𝐫1↦(−1)⁢𝐫1 to I2.

3.8.2 Doing a row operation is the same as multiplying by an elementary matrix

Doing a row operation r to a matrix has the same effect as multiplying that matrix on the left by the elementary matrix corresponding to r:

Theorem 3.8.1.

Let r be a row operation and A an m×n matrix. Then r⁢(A)=r⁢(Im)⁢A.

Proof.

We will use the fact that matrix multiplication happens rowwise. Specifically, we use Proposition 3.2.5 which says that if the rows of A are 𝐬1,…,𝐬m and if 𝐫=(a1⋯am) is a row vector then

𝐫⁢A=a1⁢𝐬1+⋯+am⁢𝐬m

and Theorem 3.2.6, which tells us that the rows of r⁢(Im)⁢A are 𝐫1⁢A, …, 𝐫m⁢A where 𝐫j is the jth row of r⁢(Im). We deal with each row operation separately.

1. 1.

   Let r be 𝐫j↦𝐫j+λ⁢𝐫i. Row j of r⁢(Im) has a 1 in position j, a λ in position i, and zero everywhere else, so by the Proposition mentioned above

   𝐫j⁢A=𝐬j+λ⁢𝐬i.

   For j′≠j, row j′ of r⁢(Im) has a 1 at position j′ and zeroes elsewhere, so

   𝐫j′⁢A=𝐬j′.

   The theorem mentioned above tells us that these are the rows of r⁢(Im)⁢A, but they are exactly the result of doing r to A.

2. 2.

   Let r be 𝐫j↦λ⁢𝐫j. Row j of r⁢(Im) has a λ in position j and zero everywhere else, so

   𝐫j⁢A=λ⁢𝐬j.

   For j′≠j, row j′ of r⁢(Im) has a 1 at position j′ and zeroes elsewhere, so

   𝐫j′⁢A=𝐬j′.

   As before, these are the rows of r⁢(Im)⁢A and they show that this is the same as the result of doing r to A.

3. 3.

   Let r be 𝐫i↔𝐫j. Row i of r⁢(Im) has a 1 in position j and zeroes elsewhere, and row j of r⁢(Im) has a 1 in position i and zeroes elsewhere, so rows i and j of r⁢(Im)⁢A are given by

   𝐫i⁢A	=𝐬j
   𝐫j⁢A	=𝐬i.

   As in the previous two cases, all other rows of r⁢(Im)⁢A are the same as the corresponding row of A. The result follows.

∎

Corollary 3.8.2.

Elementary matrices are invertible.

Proof.

Let r be a row operation, s be the inverse row operation to r, and let In an identity matrix. By Theorem 3.8.1, r⁢(In)⁢s⁢(In)=r⁢(s⁢(In)). Because s is inverse to r, this is In. Similarly, s⁢(In)⁢r⁢(In)=s⁢(r⁢(In))=In. It follows that r⁢(In) is invertible with inverse s⁢(In). ∎