How do you solve #((2, 0, 0), (-1, 2, 0), (-2, 4, 1))x=((4), (10), (11))#?

First construct the inverse matrix of #((2, 0, 0), (-1, 2, 0), (-2, 4, 1))# by writing the identity matrix alongside it and transforming that as we transform the original matrix into the identity:

#((2, 0, 0, |, 1, 0, 0), (-1, 2, 0, |, 0, 1, 0), (-2, 4, 1, |, 0, 0, 1))#

Add row #1# to row #3#:

#((2, 0, 0, |, 1, 0, 0), (-1, 2, 0, |, 0, 1, 0), (0, 4, 1, |, 1, 0, 1))#

Divide row #1# by #2#:

#((1, 0, 0, |, 1/2, 0, 0), (-1, 2, 0, |, 0, 1, 0), (0, 4, 1, |, 1, 0, 1))#

Add row #1# to row #2#:

#((1, 0, 0, |, 1/2, 0, 0), (0, 2, 0, |, 1/2, 1, 0), (0, 4, 1, |, 1, 0, 1))#

Subtract #2 xx# row #2# from row #3#:

#((1, 0, 0, |, 1/2, 0, 0), (0, 2, 0, |, 1/2, 1, 0), (0, 0, 1, |, 0, -2, 1))#

Divide row #2# by #2#:

#((1, 0, 0, |, 1/2, 0, 0), (0, 1, 0, |, 1/4, 1/2, 0), (0, 0, 1, |, 0, -2, 1))#

So:

#((2, 0, 0), (-1, 2, 0), (-2, 4, 1))^(-1) = ((1/2, 0, 0), (1/4, 1/2, 0), (0, -2, 1))#

Then multiply the right hand side column matrix by our inverse matrix to find:

#x = ((1/2, 0, 0), (1/4, 1/2, 0), (0, -2, 1))((4),(10),(11)) = ((2), (6), (-9))#

Alternatively, don't bother to construct an inverse matrix: Just perform a similar sequence of steps with the target column matrix appended to our original matrix as follows:

#((2, 0, 0, |, 4 ), (-1, 2, 0, |, 10), (-2, 4, 1, |, 11))#

Add row #1# to row #3#:

#((2, 0, 0, |, 4 ), (-1, 2, 0, |, 10), (0, 4, 1, |, 15))#

Divide row #1# by #2#:

#((1, 0, 0, |, 2 ), (-1, 2, 0, |, 10), (0, 4, 1, |, 15))#

Add row #1# to row #2#:

#((1, 0, 0, |, 2 ), (0, 2, 0, |, 12), (0, 4, 1, |, 15))#

Subtract #2 xx# row #2# from row #3#:

#((1, 0, 0, |, 2 ), (0, 2, 0, |, 12), (0, 0, 1, |, -9))#

Divide row #2# by #2#:

#((1, 0, 0, |, 2 ), (0, 1, 0, |, 6), (0, 0, 1, |, -9))#

Having reached the identity matrix on the left hand side, we can read off the solution from the right hand side:

#x = ((2), (6), (-9))#