Question

Matrix Column operations?

Answer 1

apparently you can do column operations, but i dont see how logically this holds up. row operations makes sense because you are multiplying whole equations by numbers hence why it holds the equality

Answer 2

but matrix column operations takes the first terms of several equations and then can add them to other terms, does seem to make sense. see pic

Answer 3

only elementary matrices are for row and column operations.

Answer 4

row operations are always used.... column operations aren't allowed except when you have an elementary matrix.

Answer 5

they transformed matrix A into B with column operations. I dont see how column operations make any sense.

Answer 6

then that must be an elementary matrix they are using because technically you can only use row operations

Answer 7

One reason you want to do that column operation in above context is :
If you happen to know the determinant of matrix B already and want to find the determinant of matrix A, then doing that column operation makes sense as it allows you to conclude both determinants are same.

Answer 8

That also shows that both matrix A and matrix B have same "column space", i.e, in the equation "Ax = b", the set of all vectors "b" for which the equation can be solvable is same for both matrices.

Notice column operations preserve the "column space" but row operations destroy the "column space"

As you can see what operations you do depends on what information you're seeking from the reduced matrix.