Linear Algebra – Proof of Elementary Row Operations for Matrices

linear algebramatrices

I'm taking a Linear Algebra course, and we just started talking about matrices. So we were introduced to the elementary row operations for matrices which say that we can do the following:

  1. Interchange two rows.
  2. Multiply a row with a nonzero number.
  3. Add a row to another one multiplied by a number.

Now I understood from the lecture in class how to use these and all, but I want to understand the logic behind number 3.
Is there a mathematical proof that shows that by adding row $R_1$ to row $R_2$ we are not changing the system of equations?

Thanks in advance

Best Answer

We are changing the system of equations, what we are not changing is the set of solutions to the system of equations.

The following figure shows the equations \begin{align*} -x+2y &= 2 \\ x-y &= 0 \end{align*} and what happens after we add the first equation to the second, i.e., the system of equations \begin{align*} -x+2y &= 2 \\ y &= 2. \end{align*}

enter image description here

We can see that the system of equations has indeed changed, but the set of solutions (in this case $\{(2,2)\}$) is preserved.

To prove this: If $\mathbf{x}$ satisfies equations $E_1$ and $E_2$, then it satisfies equations $E_1$ and $E_1+E_2$. Conversely, if $\mathbf{x}$ satisfies equation $E_1$ and $E_1+E_2$, then it satisfies equations $E_1$ and $(E_1+E_2)-E_1$ (which is $E_2$).

We conclude that $\mathbf{x}$ satisfies equations $E_1$ and $E_2$ if and only if $\mathbf{x}$ satisfies equations $E_1$ and $E_1+E_2$.

Thus the set of solutions is preserved.

Related Question