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Let $A=\left(\begin{array}{lll}1 & 1 & 2 \\ 1 & 1 & 2 \\ 1 & 1 & 2\end{array}\right)$.
a) What is the dimension of the image of $A ?$ Why?
b) What is the dimension of the kernel of $A ?$ Why?
c) What are the eigenvalues of $A ?$ Why?
d) What are the eigenvalues of $B:=\left(\begin{array}{lll}4 & 1 & 2 \\ 1 & 4 & 2 \\ 1 & 1 & 5\end{array}\right) ?$ Why? [HiNT: $\left.B=A+3 I\right]$.
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