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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 $-1$?

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

Siong Thye Goh
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Petras
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    Yes.$~~~~~~~~~$ – JMoravitz Nov 01 '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 01 '17 at 19:20

2 Answers2

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Yes.

We can check that it is indeed true by multiplying the two matrices together to see if we can get the identity matrix.

Multiplication of diagonal matrices $A$ and $B$ gives us another diagonal matrix $C$ where $$C_{ii}=A_{ii}B_{ii}.$$

Siong Thye Goh
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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 $\begin{pmatrix}\frac{1}{a} & 0 & 0 \\ 0 & \frac{1}{b} & 0 \\0 & 0 & \frac{1}{c}\end{pmatrix}$

user247327
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