# Thread: Determine when vector is not in range of Matrix Transformation

1. ## Determine when vector is not in range of Matrix Transformation

Let $\displaystyle R^2->R^2$be the matrix transformation defined by f(x)=Ax, where $\displaystyle A=\begin{pmatrix}2 & 1 & 2\\1 & 0 & -1\\3 & 1 & k\end{pmatrix}$

Determine k so that $\displaystyle w=\begin{pmatrix}1\\1\\1\end{pmatrix}$ is not in the range of f.

2. Originally Posted by jennifer1004
Let $\displaystyle R^2->R^2$be the matrix transformation defined by f(x)=Ax, where $\displaystyle A=\begin{pmatrix}2 & 1 & 2\\1 & 0 & -1\\3 & 1 & k\end{pmatrix}$

Determine k so that $\displaystyle w=\begin{pmatrix}1\\1\\1\end{pmatrix}$ is not in the range of f.

We want to determine k so that the column vector $\displaystyle w=\begin{pmatrix}1\\1\\1\end{pmatrix}$ is NOT in the range of f.

So, we want k such that for any vector x, $\displaystyle Ax = \begin{pmatrix}1\\1\\1\end{pmatrix}$ NEVER HOLDS.

That is, $\displaystyle \begin{pmatrix}2 & 1 & 2\\1 & 0 & -1\\3 & 1 & k\end{pmatrix} x = \begin{pmatrix}1\\1\\1\end{pmatrix}$ NEVER HOLDS.

How can we assure this? What must k be/not be?