Determine Matrix A when only the nullspace and eigenvector with eigenvalue is know.

Question

Can someone help me with the following question:

The matrix A is a 2X2-matrix. The vector $[2 , 1]^T$ is in the null space of A and $[3 , 2]^T$ is an eigenvector with eigenvalue 2. What is the matrix A?

This is an exam question

Answer 1

If $A=\left[\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right]$, then you know that:

• $A.\left[\begin{smallmatrix}2\\1\end{smallmatrix}\right]=\left[\begin{smallmatrix}0\\0\end{smallmatrix}\right]$;
• $A.\left[\begin{smallmatrix}3\\2\end{smallmatrix}\right]=\left[\begin{smallmatrix}6\\4\end{smallmatrix}\right]$.

In other words:$$\left\{\begin{array}{l}2a+b=0\\2c+d=0\\3a+2b=6\\3c+2d=4.\end{array}\right.$$Can you take it from here?

Answer 2

Sketch: You have been given sufficient information to determine a matrix $P$ such that $PAP^{-1}=\begin{pmatrix}0&0\\ 0&2\end{pmatrix}$. Once you've determined $P$ with your favourite method, $A=P^{-1}\begin{pmatrix}0&0\\ 0&2\end{pmatrix}P$.

Answer 3

You must solve

$2a+b=0$, $2c+d=0$ and $3a+2b=6$

$3c+2d=4$

So

$c=-4$, $d=8$ , $a=-6$ , $b=12$