x + y + z = 1
x + 2y + 3z = 6
x + 3y + 4z = 6
By matrix inversion.
I will assume you already know how to find the inverse
of a matrix, and how to multiply two matrices. If you don't,
post again asking how.
First we form three matrices, A, X, and B.
1. Matrix A is the 3x3 coefficient matrix A, which consists
of just the three columns of x, y, and z coefficients. in
that order, but does not contain the column of constants.
.
2. Matrix X is the 3x1 matrix of variables 
3. Matrix B is the 3x1 matrix, whose only column is the
column of constants: 
Next we form the matrix equation:
or
To solve the equation
we left-multiply both sides by 
, the inverse of 
.
Then since the associatitive principle holds for matrix multiplication,
(even though the commutative principle DOES NOT!!!), we can move
the parentheses on the left around the first two matrix factors:
Now since 
, where I is the identity matrix, the
above becomes:
and by the identity property:
Performing these operations with the actual matrices we start with
the equation 
Next we form the inverse of A, which is written A-1.
Then we indicate the left multiplication of both sides by
to get the equation 
:
Next we use the associative principle to move the parentheses so that
they are around the first two factors to get the equation 
:
Now we perform the actual multiplications and we get the equation 
:
Then when we multiply the identity matrix 
by the column matrix of
variables, we just get the matrix of variables, or the
equation 
Solution: x=1, y=-5, z=5
------------------
Checking:
x + y + z = 1
That checks.
x + 2y + 3z = 6
That checks.
x + 3y + 4z = 6
That checks.
Edwin
