I'm stuck at this question and Don't have any clue how to proceed.
The Spectral decomposition is :
$$A=\sum_{i=1}^k c_iE_i\newcommand{\adj}[1]{\mathrm{adj}\ #1}$$ As we know, the eigenvalues of $\adj{A}$ is $\frac {|A|}{c_1}$, $\ldots$, $\frac {|A|}{c_i}$ (although I don't know how to use it in this question)
I tried using the formula $\adj{A}\cdot A=|A|\cdot I$, Multiplying by $A^{-1}$ from the right and get
$$\adj{A}=|A|\cdot A^{-1} = |A|\cdot(c_1E_1+ ... + c_kE_k)^{-1}$$
And here I'm currently stuck, any hints?
Thank you very much!
From "spectral decomposition", I assume that each $E_i$ is a projection onto the eigenspace associated with eigenvalue $c_i$.
It suffices to note that $(c_1 E_1 + \cdots + c_n E_n)^{-1} = c_1^{-1}E_1 + \cdots + c_n^{-1} E_n$. You can verify that this is true by expanding the product $$ (c_1 E_1 + \cdots + c_n E_n)(c_1^{-1}E_1 + \cdots + c_n^{-1} E_n) $$ and simplify using the fact that $E_iE_j = 0$ for all $i \neq j$.