2.4 Elementary Matrices, Study notes of Elementary Mathematics

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Matrices: §2.4 Elementary Matrices

Satya Mandal, KU

Summer 2017

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Goals

I (^) Define Elementary Matrices, corresponding to elementary operations. I (^) We will see that performing an elementary row operation on a matrix A is same as multiplying A on the left by an elmentary matrix E. I (^) We will see that any matrix A is invertible if and only if it is the product of elementary matrices.

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ExampleExamples Row Equivalence Theorem 2.2Examples

Inverse of Elementary Matrices

Theorem If E is elementary, then E −^1 exists and is elementary. I (^) Proof For each of the three types of elementary matrices, write down the inverse and check. I will do it on the board.

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ExampleExamples Row Equivalence Theorem 2.2Examples

Example 2.4.

Let

A =

Is this matrix elementary. If yes why? Answer: Yes, it is. The matrix A is obtained from I 3 by adding 3 time the first row of I 3 to the second row.

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ExampleExamples Row Equivalence Theorem 2.2Examples

Example 2.4.

Let

A =

Is this matrix elementary. If yes why? Answer: Yes, it is. The matrix A is obtained from I 3 by switching its first and third row.

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ExampleExamples Row Equivalence Theorem 2.2Examples

Theorem 2.

Theorem. Let A be a matrix of size m × n. Let E be an elementary matrix (of size m × m) obtained by performing an elementary row operation on Im and B be the matrix obtained from A by performing the same operation on A. Then B = EA.

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ExampleExamples Row Equivalence Theorem 2.2Examples

Example 2.4.

Let

A =

 , B =

Find an elementary matrix E so that B = EA. Solution: The matrix B is obtained by switching first and the last row of A. They have size 3 × 4. By the theorem above, E is obtained by switching first and the last row of I 3. So,

E =

 (^) , so B = EA (Directly Check, as well.).

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ExampleExamples Row Equivalence Theorem 2.2Examples

Example 2.4.

Let

A =

 , B =

Find an elementary matrix E so that B = EA. Solution: The matrix B is obtained by adding 2 times the first row of A to the second row of A. By the thorem above, E is obtained from I 3 by adding 2 times its first row to second. So,

E =

 (^) , so B = EA (Directly Check, as well.).

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ExampleExamples Row Equivalence Theorem 2.2Examples

Definition

Definition. Two matrices A, B of size m × n are said to be row-equivalent if

B = Ek Ek− 1 · · · E 2 E 1 A where Ei are elemetary.

This is same as saying that B is obtained from A by application of a series of elemetary row operations.

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ExampleExamples Row Equivalence Theorem 2.2Examples

Theorem 2.

Theorem. A square matrix A is invertible if and only if it is product of elementary matrices. Proof. Need to prove two statements. First prove, if A is product it of elementary matrices, then A is invertible. So, suppose A = Ek Ek− 1 · · · E 2 E 1 where Ei are elementary. Since elementary matrices are invertible, E (^) i− 1 exists. Write B = E 1 − 1 E 2 − 1 · · · E (^) k−−^11 E (^) k− 1. Then

AB = (Ek Ek− 1 · · · E 2 E 1 )(E 1 − 1 E 2 − 1 · · · E (^) k−−^11 E (^) k− 1 ) = I.

Similarly, BA = I. So, B is the inverse of A.

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ExampleExamples Row Equivalence Theorem 2.2Examples

Example 2.4.

Let

A =

Find its inverse, using the theorem above. Solution. The method is to reduce A to I 3 by elementary operations, and interpret it in terms of multiplication by elementary matrices.

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ExampleExamples Row Equivalence Theorem 2.2Examples

Continued

First, subtract 2 time the first row from second, which is same as multiplying A by the elementary matrix

E 1 =

 (^). So E 1 A =

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ExampleExamples Row Equivalence Theorem 2.2Examples

Continued

Now, multiply the first row of E 3 E 2 E 1 A by .5. So, with

E 4 =

 , E 4 E 3 E 2 E 1 A =

 = I.

So, A = E 1 − 1 E 2 − 1 E 3 − 1 E 4 −^1 If you wish, you can write it more explicitly, by expanding the right hand side.

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Example 2.4.

Let A =

 , C =

Find an elementary matrix so that EA = C. Solution. If we add third row of A to its first row, we get C. Let E be the matrix that is obtained from the identity matrix I 3 by adding its third row to the first. Or

E =

 (^) , so EA = C.