Question

Find all 2x2matrices for which $\left[\begin{array}{c}2\\ 3\end{array}\right]$is an eigenvector with associated eigenvalue -1.

Short answer

So, the matrix is $\left\{\left[\begin{array}{c}a\frac{-2-2a}{3}\\ 3\frac{-3-2c}{3}\end{array}\right]\mathrm{Ia},\mathrm{c}\in \mathrm{ℝ}\right\}$for which $\left[\begin{array}{c}2\\ 3\end{array}\right]$ is an eigenvector with associated eigenvalue -1.

Step-by-step solution

 Step 1: Define the eigenvector

Eigenvector:An eigenvector of Ais a nonzero vector vin${\mathbf{R}}_{\mathbf{n}}$such that $\mathbf{Av}\mathbf{=}\mathbf{\lambda v}$, for some scalar$\mathbf{\lambda }$.

Solve for A

If $\stackrel{\rightharpoonup }{v}$is an Eigen vector of A this means that:

$A\stackrel{\rightharpoonup }{v}=\lambda \text{'}v$

Now let $\mathrm{A}=\left[\begin{array}{cc}\mathrm{a}& \mathrm{b}\\ \mathrm{c}& \mathrm{d}\end{array}\right],\stackrel{\rightharpoonup }{\mathrm{v}}=\mathrm{and}\mathrm{\lambda }=-1$and

We want to solve for A.

Evaluation

By the definition of eigenvector,

$$\begin{gathered} A\stackrel{\rightharpoonup }{v}=\lambda \stackrel{\rightharpoonup }{v} \\ \left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{c}2\\ 3\end{array}\right]=-1\left[\begin{array}{c}2\\ 3\end{array}\right] \\ \left[\begin{array}{c}2a+3b\\ 2c+3d\end{array}\right]=\left[\begin{array}{c}-2\\ -3\end{array}\right] \\ \mathrm{By}\mathrm{the}\mathrm{equality}\mathrm{of}\mathrm{matrices}, \\ \begin{array}{c}2\mathrm{a}+3\mathrm{b}=-2\\ 2\mathrm{c}+3\mathrm{d}=-3\end{array} \end{gathered}$$

Solving the equations

Solve the equations 2a+3b=-2 and 2c+3d=-3d to get $b=\frac{-2-2a}{3}$and $d=\frac{-3-2c}{3}$.

Thus, the form of the matrix for which the vector $\left[\begin{array}{c}2\\ 3\end{array}\right]$is an eigenvector with associated eigenvalue -1 is $\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=\left[\begin{array}{cc}a& \frac{-2-2a}{3}\\ c& \frac{-3-2c}{3}\end{array}\right]$.

Therefore, the required matrices is $\left\{\left[\begin{array}{cc}a& \frac{-2-2a}{3}\\ c& \frac{-3-2a}{3}\end{array}\right]\mathrm{Ia},\mathrm{c}\in \mathrm{ℝ}\right\}$