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True or False - and Why?
a) A $3 \times 3$ real matrix need not have any real eigenvalues.
b) If an $n \times n$ matrix $A$ is invertible, then it is diagonalizable.
c) If $A$ is a $2 \times 2$ matrix both of whose eigenvalues are 1 , then $A$ is the identity matrix.
d) If $\vec{v}$ is an eigenvector of the matrix $A$, then it is also an eigenvector of the matrix $B:=A+7 I$.
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a) True. A $3 \times 3$ real matrix can have complex eigenvalues, which are not real numbers.

b) False. An $n \times n$ matrix $A$ can be invertible but not diagonalizable. For example, the Jordan form of a matrix can be invertible but not diagonalizable.

c) False. If an $2 \times 2$ matrix has both eigenvalues equal to 1 , it does not necessarily mean that it is the identity matrix. For example, the matrix:
$\begin{bmatrix} 2 & 1 \\ 0 & 2 \end{bmatrix}$

has both eigenvalues equal to 1 , but it is not the identity matrix.

d) True. If $A$ is a square matrix and $\vec{V}$ is an eigenvector of $A$ with eigenvalue $\lambda$, then: $A\vec{v} = \lambda;\vec{v}$

Adding a multiple of the identity matrix to both sides of the above equation yields:

$(A + 7I)\vec{v} = (\lambda; + 7)\vec{v}$
This shows that if $A$ has an eigenvector with eigenvalue $\lambda$, then $A+7I$ has an eigenvector with eigenvalue $\lambda+7$
by Platinum (101k points)

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