Obtaining the values of $a$ and $b$ given their eigenvector and matrix

Question

Let $A$ be $3×3$ matrix as:

$$A=\begin{pmatrix} 2 & 1 & 1\\ a & 3 & 2\\ 3 & b & c \end{pmatrix}$$

It is required to obtain the Values of $a$, $b$ and $c$ if the Eigen vectors of the given matrix are $(1,0,-1)$ and $(0,1,-1)$.

My approach

I used the following equation to solve for the values of $a$ and $b$:

$$\begin{pmatrix} 2 & 1 & 1\\ a & 3 & 2\\ 3 & b & c \end{pmatrix}\begin{pmatrix} 1\\ 0\\ -1 \end{pmatrix}= \begin{pmatrix} 1\\ 0\\ -1 \end{pmatrix}$$

Doing the same with the second Eigen vector and after solving the two equations, we get the values as $a=2$ , $b=3$ and $c=4$.

It looks correct to me although I am a little skeptical about the above relationship between matrix and Eigen vectors.

Thanks.

Answer 1

Your approach is correct.

We just have to multiply the matrix $A$ with the eigenvector $v_i$, and from the equation $$Av_i = \lambda_iv_i,$$

we can identify the value of $\lambda_i=1$.

From the first eigenvector discover the value of $a$ and $c$ and then from the second eigencector discover the value of $b$.

To be more detailed, for example, for the first eigenvector,

$$\begin{pmatrix} 2 & 1 & 1\\ a & 3 & 2\\ 3 & b & c \end{pmatrix}\begin{pmatrix} 1\\ 0\\ -1 \end{pmatrix}=\begin{pmatrix} 1\\ a-2\\ 3-c \end{pmatrix}= \lambda_1 \begin{pmatrix} 1\\ 0\\ -1 \end{pmatrix}.$$

From the first entry, we conclude that $\lambda_1 = 1$. After which we solve for $a-2=0$ and $3-c=-1$.