10. Solve using row operations
We first look at the augmented matrix.
We perform the same operations we as we do when we are trying to invert the coefficient matrix. First we need to get a 1 in the upper left corner. The easiest way to do that would be to switch the first two rows.
We can now use the 1 in the upper left corner to get rid of the other entries in the first column. Multiply the top row by 2 and subtract it from the second row to get a new second row.
and multiply it by 4 ans subtract it from the tihird row to get a new third row.
Our matrix now looks like
We next want to get a 1 in the second column of the second row. The easiest way to do that would be to start by dividing the third row by 3.
This gets us a 1 in the second column. If we switch the second and third rows, we get a 1 in the desired position.
We can now use this 1 to wipe out all of the other entries in the second column. To get rid of the second entry in the top row, add the two rows to get a new top row.
and subtract 5 times the middle row from thebottom row to get rid of the 5.
Our matrix now looks like
We next want to get a 1 in the third column of the third row. In order to do that we will have to divide the third row by 14.
We can now use this 1 to wipe out the other entries in the third column. First we multiply the new bottom row by 2 and add it to the top row to get rid if the -2.
and we multiply the bottom row by 3 and add it to the middle row to get rid of the -3.
Our matrix now looks like
which corresponds to the system of equations
which is the same solution we get from substitution, addition, Cramer's rule, and the inverse method, and which checks.