Question

Factor the matrix A=[[2,-4],[1,0]] into a product of elementary matrices.

Answer 1

To factor the matrix A=[[2,-4],[1,0]] into a product of elementary matrices, we need to perform row operations to transform A into the identity matrix. The elementary matrices corresponding to the row operations can then be multiplied together to obtain the factorization. The resulting factorization is (1/2) * [[1,0],[0,1]] * [[2,-4],[1,2]] * [[1/2,0],[-1/2,1]].

Step-by-step explanation:

To factor the matrix A=[[2,-4],[1,0]] into a product of elementary matrices, we can use row operations. The goal is to transform matrix A into the identity matrix I. We can achieve this by performing the same row operations on the identity matrix as we do on matrix A. The elementary matrices corresponding to each row operation can then be multiplied together to obtain the desired factorization. Let's go step by step:

1. Begin with the matrix A and the identity matrix I: A=[[2,-4],[1,0]], I=[[1,0],[0,1]].
2. Perform the following row operations to transform A into I:
3. 1. Multiply the first row of A by 1/2.
4. 2. Add -1 times the first row of A to the second row of A.
5. Perform the same row operations on the identity matrix I:
6. 1. Multiply the first row of I by 1/2.
7. 2. Add -1 times the first row of I to the second row of I.
8. Multiply the resulting elementary matrices together to obtain the factorization of A.

The resulting factorization is:

A = (1/2) * [[1,0],[0,1]] * [[2,-4],[1,2]] * [[1/2,0],[-1/2,1]].