How to find $\lambda$ in this situation? Inverse of a $2\times1$ matrix?

Question

I was solving a question to find $\lambda$ and this is the current situation:

$$\begin{bmatrix} 5 & 10\\ 15 & 10 \end{bmatrix} =\lambda\begin{bmatrix}2\\3\end{bmatrix}$$

My algebra suggested that I should divide the right-hand matrix with the left-hand one to find $\lambda$. Then after a few searches I found out that $A/B = AB^{-1}$, so now I think I need to find the inverse of that $2\times1$ matrix to multiply it with the other matrix to find $\lambda$, but the inverse is only present for a square matrix. I also have a feeling that this is not the right approach, so I am here for help. Basically I have no idea how to find $\lambda$.

Answer 1

The question, as stated, is malformed. On the left you have a 2x2 matrix; on the right, a 2x1. The two cannot be equal under any circumstances. This tells me that either there was a typo in the question or you simply misread it.

If you're looking at eigenvalues and eigenvectors, then the correct formulation of the question is almost certainly: $ \left(\begin{array}{cc} 5 & 10\\ 15 & 10 \end{array}\right) \left(\begin{array}{c} 2\\ 3 \end{array}\right) = \lambda \left(\begin{array}{c} 2\\ 3 \end{array}\right) $. If this is the correct question, then the approach is to multiply the left side out, to get $\left(\begin{array}{c} 40\\ 60 \end{array}\right) = \lambda\left(\begin{array}{c} 2\\ 3 \end{array}\right)$. Then break this into two equations, equating each component: $40 = 2\lambda$ and $60 = 3\lambda$. You can now solve for $\lambda$.

Answer 2

An eigenvector $V$ of the matrix $A$ is a vector such that $AV$=$ \lambda V$.

So what you want to do is work out $AV= (40, 60)^T=\lambda (2,3)^T$ from this we see that $\lambda=20$. This is from the definition of eigenvectors and eigenvalues.