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I have a problem of...

Let $A$ be a 2x2 matrix such that it is not invertible and 2 is an eigenvalue of $A$.

a) Find all eigenvalues of $A+I$.

b) Prove or disprove A+I is invertible.

Since it's not invertible, it has an eigenvalue of 0. So I can think of a matrix easily such as the one below with eigenvalues of 2 and 0...

$\begin{bmatrix} 0 & a \\ 0 & 2 \end{bmatrix}$

Where $a$ is just some unknown. However, I'm assuming there are many matrices that have eigenvalues of 2 and 0 for a 2x2. So I am having trouble even seeing what the eigenvalues will even tell me about the original matrix.

Do eigenvalues tell you anything about the original structure of the matrix?

Also I haven't learned about eigenvectors yet in class.

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You don't have to find the original matrix to answer the question. As you said the eigenvalues must be $\lambda =0,2$. Claim, $\mu =1,3$ are the eigenvalues of $A+I$. Lets check, Let $x$ be an eigenvector of $A$ corresponding to the eigenvalue 2, then $$(A+I)x=Ax+x=2x+x=3x.$$Thus, $\mu=3$ is an eigenvalue for $A+I$. Similarly, we can conclude that $\mu=1$ is the other eigenvalue. Which means $A+I$ is invertible.

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    $\begingroup$ You do need to mention for the final conclusion that a $2\times2$ matrix cannot have more than $2$ distinct eigenvalues, so with $1$ and $3$ already taken, $0$ cannot be an eigenvalue of $A+I$. $\endgroup$ – Marc van Leeuwen Oct 18 '16 at 6:18
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The Given matrix has two distinct eigenvalues $0$ and $2$. Therefore it will have $2$ linearly independent eigenvectors and will be diagonalizable. That is, it will be similar to $$ \begin{bmatrix} 2 & 0 \\ 0 & 0 \\ \end{bmatrix} $$ So we can right $A+I$ as \begin{bmatrix} 3 & 0 \\ 0 & 1 \\ \end{bmatrix} which is a invertible matrix with eigenvalues 3 and 1.

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