Question

How do you find the inverse of `A=``((2, 4, 1),(-1, 1, -1), (1, 4, 0))`?

Answer 1

The inverted matrix is: `((-4,-4,5),(1,1,-1),(5,4,-6))`

Explanation:

There are many ways in invert matrices, but for this problem I used the cofactor transpose method.

If we imagine that
`A = ((vecA),(vecB),(vecC))`

So that:
`vecA = (2,4,1)`
`vecB = (-1,1,-1)`
`vecC = (1,4,0)`

Then we can define reciprocal vectors:
`vecA_R = vecB xx vecC`
`vecB_R = vecC xx vecA`
`vecC_R = vecA xx vecB`

Each is easily calculated using the determinant rule for cross products:
`vecA_R = |(hati,hatj,hatk),(-1,1,-1),(1,4,0)| = (4,-1,-5)`
`vecB_R = |(hati,hatj,hatk),(-1,4,0),(2,4,1)| = (4,-1,-4)`
`vecC_R = |(hati,hatj,hatk),(2,4,1),(-1,1,-1)| = (-5,1,6)`

We can use these to construct the cofactor transpose of `M`, `barM`, as such:
`barM = ((vecA_R^T,vecB_R^T,vecC_R^T)) = ((4,4,-5),(-1,-1,1),(-5,-4,6))`

The reciprocal vectors and the cofactor transpose matrix have two interesting properties:
`vecA*vecA_R=vecB*vecB_R=vecC*vecC_R = det(M)`
and
`M^-1 = barM/detM`

So we can determine that:
`det(M) = vecC*vecC_R = (1,4,0)*(-5,1,6) = -1`

This means that:
`M^-1 = -barM/1 = -((4,4,-5),(-1,-1,1),(-5,-4,6)) = ((-4,-4,5),(1,1,-1),(5,4,-6))`