How do you solve #2x-3y+z=-2#, #x-6y+3z=-2# and #3x+3y-2z=2# using matrices?

One way to do this is to use the Gauss-Jordan Method.

Replace the given equations with an augmented matrix
Initial Augmented Matrix:
#( (color(white)("XXX")x,color(white)("XXX")y,color(white)("XXXX")z,color(white)("XXX")=), (color(white)("XXX")2.0,color(white)("XXXX")-3.0,color(white)("XXX")1.0,color(white)("XXXX")-2.0), (color(white)("XXX")1.0,color(white)("XXXX")-6.0,color(white)("XXX")3.0,color(white)("XXXX")-2.0), (color(white)("XXX")3.0,color(white)("XXX")3.0,color(white)("XXXX")-2.0,color(white)("XXX")2.0) )#

Pivot Row: 1 Pivot Column: 1
pivot Row reduced:
#( (color(white)("XXX")1.0,color(white)("XXXX")-1.5,color(white)("XXX")0.5,color(white)("XXXX")-1.0), (color(white)("XXX")1.0,color(white)("XXXX")-6.0,color(white)("XXX")3.0,color(white)("XXXX")-2.0), (color(white)("XXX")3.0,color(white)("XXX")3.0,color(white)("XXXX")-2.0,color(white)("XXX")2.0) )#

Non-pivot rows reduced for pivot column
#( (color(white)("XXX")1.0,color(white)("XXXX")-1.5,color(white)("XXX")0.5,color(white)("XXXX")-1.0), (color(white)("XXX")0.0,color(white)("XXXX")-4.5,color(white)("XXX")2.5,color(white)("XXXX")-1.0), (color(white)("XXX")0.0,color(white)("XXX")7.5,color(white)("XXXX")-3.5,color(white)("XXX")5.0) )#

Pivot Row: 2 Pivot Column: 2
pivot Row reduced:
#( (color(white)("XXX")1.0,color(white)("XXXX")-1.5,color(white)("XXX")0.5,color(white)("XXXX")-1.0), (color(white)("XXX")0.0,color(white)("XXX")1.0,color(white)("XXXX")-0.556,color(white)("XXX")0.222), (color(white)("XXX")0.0,color(white)("XXX")7.5,color(white)("XXXX")-3.5,color(white)("XXX")5.0), (color(white)("XXX")x,color(white)("XXX")y,color(white)("XXXX")-z,color(white)("XXX")=) )#

Non-pivot rows reduced for pivot column
#( (color(white)("XXX")1.0,color(white)("XXX")0.0,color(white)("XXXX")-0.333,color(white)("XXXX")-0.667), (color(white)("XXX")0.0,color(white)("XXX")1.0,color(white)("XXXX")-0.556,color(white)("XXX")0.222), (color(white)("XXX")0.0,color(white)("XXX")0.0,color(white)("XXX")0.667,color(white)("XXX")3.333) )#

Pivot Row: 3 Pivot Column: 3
pivot Row reduced:
#( (color(white)("XXX")1.0,color(white)("XXX")0.0,color(white)("XXXX")-0.333,color(white)("XXXX")-0.667), (color(white)("XXX")0.0,color(white)("XXX")1.0,color(white)("XXXX")-0.556,color(white)("XXX")0.222), (color(white)("XXX")0.0,color(white)("XXX")0.0,color(white)("XXX")1.0,color(white)("XXX")5.0) )#

Non-pivot rows reduced for pivot column
#( (color(white)("XXX")1.0,color(white)("XXX")0.0,color(white)("XXX")0.0,color(white)("XXX")1.0), (color(white)("XXX")0.0,color(white)("XXX")1.0,color(white)("XXX")0.0,color(white)("XXX")3.0), (color(white)("XXX")0.0,color(white)("XXX")0.0,color(white)("XXX")1.0,color(white)("XXX")5.0), (color(white)("XXX")x,color(white)("XXX")y,color(white)("XXXX")z,color(white)("XXX")=) )#