the product of two n x n elementary matrices in general is not necessarily an elementary matrix
1
An elementary matrix is a matrix obtained from an identity matrix by performing a single elementary row operation (such as swapping two rows, scaling a row by a nonzero constant, or adding a multiple of one row to another).
Suppose we have two n x n elementary matrices E and F. Then, the product EF is obtained by performing the row operation corresponding to F on the identity matrix, and then performing the row operation corresponding to E on the resulting matrix.
It is not necessarily true that the result of these two row operations will be another elementary matrix.
For example, let E be the elementary matrix obtained by adding 2 times the first row to the second row of the 3 x 3 identity matrix. Let F be the elementary matrix obtained by scaling the second row of the 3 x 3 identity matrix by a factor of 3. Then, the product EF is:
EF = (scaling the second row of I by 3) x (adding 2 times the first row to the second row of the result)
EF = [1 0 0; 0 3 0; 0 0 1] x [1 2 0; 0 1 0; 0 0 1]
EF = [1 2 0; 0 3 0; 0 0 1]
The matrix EF is not an elementary matrix because it cannot be obtained from the identity matrix by performing a single elementary row operation. Therefore, the statement the product of two n x n elementary matrices in general is not necessarily an elementary matrix is true.
More Answers:
Matrix Addition For Linear Transformations: Proof That [T+U] = [T]+[U] With Given Ordered BasesElementary Column Operations: How They Simplify And Transform Matrices Without Affecting Determinant And Eigenvalues
Simplifying Determinants Of Matrices: How To Calculate Determinants Of Transpose And Higher Powers Of A Matrix