Question
Prove that a diagonal matrix $D=\operatorname{diag}\left(d_1, \ldots, d_n\right)$ is invertible if and only if all its diagonal entries are nonzero, in which case $D^{-1}=\operatorname{diag}\left(1 / d_1, \ldots, 1 / d_n\right)$.
Step 1
This means that $D$ is an $n \times n$ matrix where the entries $D_{ii} = d_i$ for $i = 1, \ldots, n$, and all off-diagonal entries are zero, i.e., $D_{ij} = 0$ for $i \neq j$. Show more…
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