Question

Suppose a matrix B' is obtained from a matrix B my multiplying the j-th row of B by a non-zero scalar k. Prove that Row(B) = Row(B')

Answer 1

the rows represent coeffs of equations

2x + 3y = 5 for example creates a line on the graph

n(2x + 3y = 5)
2nx + 3ny = 5n represents the same equation scaled by some number


if we use say, elimination, techniques we should get the result that 0=0

2x + 3y = 5 ; *-n
2nx + 3ny = 5n

-2nx -3ny =-5n
2nx+3ny = 5n
---------------
0 + 0 = 0

in other words, they are the same line and still retain all the same points.

Answer 2

scaling an equation alters its "form", not its intrinsic values