Please check my work on this Augmented matrix problem.

[shivers20]

[ 1 -4  5  0  7 ]
[ 0  1 -3  0  6 ]
[ 0  0  1  0  2 ]
[ 0  0  0  1 -5 ]

State the next two elementary row operations that should be performed in the process of solving the system.

My answer: Equal -4 in row 1 to 0 by multiplying row 2 by 4 and sum of row1,

[ 1  0 -7  0  31 ]
[ 0  1 -3  0   6 ]
[ 0  0  1  0   2 ]
[ 0  0  0  1  -5 ]

Then my second operation would be to make the -3 in row 2 into a zero. Am I correct? There are different ways/order of approaching this problem right?

 

[soroban]

Hello, shivers20!

\(\displaystyle \begin{pmatrix} 1 &-4 & 5 & 0 & | & 7 \\ 0 & 1 &-3 & 0 & | & 6 \\ 0 & 0 & 1 & 0 & | & 2 \\ 0 & 0 & 0 & 1 & | &-5\end{pmatrix}\)

State the next two elementary row operations that should be performed in the process of solving the system.

My answer: Equal -4 in row 1 to 0 by multiplying row 2 by 4 and sum of row1,

\(\displaystyle R1\,+\,4\cdot R2\;\;\begin{pmatrix} 1 & 0 &-7 & 0 & | & 31 \\ 0 & 1 & -3 & 0 & | & 6 \\ 0 & 0 & 1 & 0 & | & 2 \\ 0 & 0 & 0 & 1 & | & -5 \end{pmatrix}\) . . . Correct!

Then my second operation would be to make the -3 in row 2 into a zero. Am I correct? . . . Yes!

There are different ways/order of approaching this problem, right? . . . Right!

Your way would be: \(\displaystyle \,R2\,+\,3\cdot R3\;\;\begin{pmatrix}1 & 0 & -7 & 0 & | & 31 \\ 0 & 1 & 0 & 0 & | & 12 \\ 0 & 0 & 1 & 0 & | & 2 \\ 0 & 0 & 0 & 1 & | & -5\end{pmatrix}\) . . . followed by the move given below.

Another way: \(\displaystyle \:R1\,+\,7\cdot R3\;\;\begin{pmatrix}1 & 0 & 0 & 0 & | & 45 \\ 0 & 1 & -3 & 0 & | & 6 \\ 0 & 0 & 1 & 0 & | & 2 \\ 0 & 0 & 0 & 1 & | & -5\end{pmatrix}\) . . . followed by your move.