# How does the inverse of diagonal matrix looks like? [closed]

How does the inverse diagonal matrix looks like - D$(3,3)$?

If I have diagonal matrix like this:

$$\begin{bmatrix} 5 & 0 & 0 \\ 0 & 7 & 0 \\ 0 & 0 & 6\end{bmatrix}$$

Is the inverse of this matrix is all non zero element raised by power of $-1$?

$$\begin{bmatrix} \frac15 & 0 & 0 \\ 0 & \frac17 & 0 \\ 0 & 0 & \frac16\end{bmatrix}$$

## closed as off-topic by José Carlos Santos, 5xum, Ben, Claude Leibovici, mechanodroidNov 2 '17 at 12:23

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• Yes.$~~~~~~~~~$ – JMoravitz Nov 1 '17 at 19:14
• As the others have said, yes. However, remember that this does not hold for all matrices. In general, you cannot just raise each element to the power of $-1$. But it is true for diagonal matrices. – Eff Nov 1 '17 at 19:20

Multiplication of diagonal matrices $A$ and $B$ gives us another diagonal matrix $C$ where $$C_{ii}=A_{ii}B_{ii}.$$
Yes, the multiplicative inverse of a diagonal matrix is a diagonal matrix with the reciprocal of the diagonal numbers on its diagonal. The multiplicative inverse of $\begin{pmatrix}a & 0 & 0 \\ 0 & b & 0 \\0 & 0 & c\end{pmatrix}$ is $\begin{pmatrix}\frac{1}{a} & 0 & 0 \\ 0 & \frac{1}{b} & 0 \\0 & 0 & \frac{1}{c}\end{pmatrix}$