the sum of two n x n elementary matrices is still an elementary matrix
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Let’s first define what an elementary matrix is. An n x n matrix is said to be an elementary matrix if it can be obtained by performing a single elementary row operation on the n x n identity matrix.
There are three types of elementary row operations:
1. Swapping two rows
2. Multiplying a row by a nonzero constant
3. Adding a multiple of one row to another row
Now, consider two n x n elementary matrices, E and F, that are obtained by performing elementary row operations on the n x n identity matrix. The sum of these matrices is given by:
(E + F) = I + A + B
where A and B are obtained by performing the same elementary row operations on the n x n zero matrix. Therefore, A and B are themselves elementary matrices.
Now, let’s consider the three types of elementary row operations:
1. Swapping two rows: If E and F are obtained by swapping two different rows of the identity matrix, then their sum (E + F) is also an elementary matrix obtained by swapping the same two rows on the identity matrix.
2. Multiplying a row by a nonzero constant: If E and F are obtained by multiplying a row of the identity matrix by a nonzero constant, then their sum (E + F) is also an elementary matrix obtained by multiplying the same row of the identity matrix by the sum of the two constants.
3. Adding a multiple of one row to another row: If E and F are obtained by adding a multiple of one row to another row of the identity matrix, then their sum (E + F) is also an elementary matrix obtained by adding the same multiple of one row to another row of the identity matrix.
Therefore, we have shown that the sum of two n x n elementary matrices is still an elementary matrix.
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