Question

How do you solve `3y-z=-1`, `x+5y-z=-4`, `-3x+6y+2z=11` using matrices?

Answer 1

Use Cramer's Rule (with Determinants) to get
`color(white)("XXX")(x,y,z)=(-3,0,1)`

Explanation:

Using the coefficient of `x, y, and z` plus the equated constants as columns we can write these equations in matrix form.

If `D` is the determinant of the variable coefficient matrix
and `D_a, ain{x,y,z}` is the determinant of the variable coefficient matrix with the column for variable `a` replaced by the equated constants column,

Cramer's Rule tells us that:
`color(white)("XXX")a=D_a/D` for `ain{x,y,z}`

An interesting point to observe, because of the way computers perform (what is called "floating point") arithmetic the value of `D_y` shows up as being
`color(white)("XXX")3E-16 = 0.0000000000000003` instead of `0`
which, in turn, cause `y` to display as
`color(white)("XXX")-1.9E-17=-0.000000000000000019` instead of `0`

You must be prepared to apply reasonable interpretations when working with computer generated outputs.