MHB Admiral Ackbar's question at Yahoo Answers (Inverse image of a vector)

[Fernando Revilla]

Here is the question:

Could someone explain this? I need to know it for a test, so it would be great if anyone could help.

A linear transformation T: ℝ^3 --> ℝ^3 has matrix
A =
[ 1 -3 1 ]
[ 2 -8 8 ]
[-6 3 -15 ]
Find a vector v in ℝ^3 that satisfies T(v) = [4 -2 9]^T .

Here is a link to the question:

Find vector that satisfies the linear transformation, linear algebra question, PLEASE HELP? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.

 

[Fernando Revilla]

Hello Admiral Ackbar,

The determinant of the given matrix $A$ is $\det A=108$, so is invertible. We have $$Av=\begin{bmatrix}{4}\\{-2}\\{9}\end{bmatrix}\Leftrightarrow v=A^{-1}\begin{bmatrix}{4}\\{-2}\\{9}\end{bmatrix}=\begin{bmatrix}{1}&{-3}&{1}\\{2}&{-8}&{8}\\{-6}&{3}&{-15}\end{bmatrix}^{-1}\begin{bmatrix}{4}\\{-2}\\{9}\end{bmatrix}=\ldots$$