Finding elementary matrices E1 and E2 such that: B = E1E2A, confused

E1 = 1 01 1E2 = 1 00 -3This will cause the second row to be added to the first, and then the second row multiplied by -3. The result will be:E2E1A = 1 00 -3E1A = 2 31 11A = 2 3-1 8Therefore, in summary, the task is to find two elementary matrices, E1 and E2, such that when applied to matrix A in the given order, will result in matrix B. The two row operations needed are adding the second
  • #1
mr_coffee
1,629
1
Hello everyone, I've been searching in this book forever to find an example but no luck. My problem states:
Find two Elementary matrices E1 and E2 such that [tex]B = E_2E_1A[/tex]
A =
2 3
-1 8

B =
1 11
3 -24

Can someone explain to me what they want me to do?
The books says:
A -> E1A->E2E1A -> Ek ... E2E1A;
Ei denote the elementary matrix corresponding to the ith row operation. and suppose that a series of k elementary row operations is applied to an arbitrary matrix A.

THanks!
 
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  • #2
mr_coffee said:
Hello everyone, I've been searching in this book forever to find an example but no luck. My problem states:
Find two Elementary matrices E1 and E2 such that [tex]B = E_2E_1A[/tex]
A =
2 3
-1 8

B =
1 11
3 -24

Can someone explain to me what they want me to do?

Staring at the matrices for a moment makes it clear that the two row operations they want are:

(1) Add the second row to the first row.
(2) Multiply the second row by -3.

Can you write down these matrices?

Carl
 

1. How do I find elementary matrices E1 and E2 to solve the equation B = E1E2A?

To find the elementary matrices E1 and E2, you can use the row operations method. Start by writing out the given matrix A and the desired result matrix B. Then, perform row operations on A until it is transformed into B. The row operations performed will correspond to the elementary matrices E1 and E2.

2. What is the purpose of using elementary matrices in this equation?

Elementary matrices are used to simplify the process of solving equations and performing row operations on matrices. By using elementary matrices, you can easily manipulate the rows or columns of a matrix without changing its properties, making it easier to find the desired result.

3. Can I use any elementary matrices to solve this equation?

Yes, you can use any elementary matrices as long as they correspond to the row operations performed on matrix A to transform it into matrix B. However, it is important to note that the order in which the elementary matrices are multiplied matters, so make sure to follow the correct order.

4. What is the difference between E1 and E2 in this equation?

E1 and E2 represent two separate elementary matrices that are multiplied together to get the desired result B. E1 corresponds to the elementary matrix used in the first step of row operations, while E2 corresponds to the elementary matrix used in the second step.

5. Is there a specific way to represent elementary matrices?

Elementary matrices are typically represented in the form of an identity matrix with one row or column being modified by a single row operation. This modified row or column will contain the coefficients of the row operation, while the rest of the elements in the identity matrix will be zeros.

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