Decomposing Matrices into Elementary Matrices: A Reverse Approach

In summary, the conversation is about writing a matrix A as a product of elementary matrices. The question is asking for the method to do this, which is to write A-1 as a product of elementary matrices and then take the inverse of that. The question asks if this is the same as writing A-1 backwards, which is correct.
  • #1
_Steve_
19
0
I have a question about elementary matrices,
I have matrix A, and I just found A-1, and then the question wants me to write A-1 as a product of elementary matrices.
Ok, that's easy, but now the question wants me to write A as a product of elementary matrices, how do I go about doing this?
Would it be the same as writing A-1 as a product of elementary matrices but backwards?
 
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  • #2
lol does that question make sense? or should I describe it better?
 
  • #3
If A-1=E3E2E1, then A equals the inverse of that; that is A=(E3E2E1)-1=E1-1E2-1E3-1.
 

1. What is an elementary matrix?

An elementary matrix is a square matrix that is obtained by performing a single elementary row operation on the identity matrix. It is used in matrix operations to simplify calculations and solve systems of linear equations.

2. What are the three types of elementary row operations?

The three types of elementary row operations are: multiplying a row by a non-zero constant, adding a multiple of one row to another row, and swapping two rows. These operations can be used to manipulate a matrix without changing its solution set.

3. How do elementary matrices help with solving systems of linear equations?

Elementary matrices can be used to transform a system of linear equations into an equivalent system with a simpler form, making it easier to solve. By multiplying the original matrix by the appropriate elementary matrix, the system can be reduced to a triangular form, making it easier to find the solution.

4. Can elementary matrices be used for operations other than solving systems of linear equations?

Yes, elementary matrices can also be used for other operations such as finding inverses of matrices, computing determinants, and solving eigenvalue problems. They are a useful tool in linear algebra and have various applications in fields such as engineering and computer science.

5. How are elementary matrices related to elementary row operations?

Elementary matrices are created by performing elementary row operations on the identity matrix. This means that the inverse of an elementary matrix is also an elementary matrix, and the product of two elementary matrices is also an elementary matrix. Therefore, elementary matrices provide a convenient way to represent and perform elementary row operations on matrices.

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