If `D=diag(d_1,d_2,d_3,…,d_n)" where "d ne 0" for all " I = 1,2,…,n," then " D^(-1)`is equal to

Options

A. 0
B. `I_n`
C. diagonal `(d_1^(-1), d_2^(-1),….d_n^(-1))`
D. None of above

Answer

To find the inverse of the diagonal matrix \( D = \text{diag}(d_1, d_2, d_3, \ldots, d_n) \) where \( d_i \neq 0 \) for all \( i = 1, 2, \ldots, n \), we can follow these steps:

### Step 1: Understand the structure of the diagonal matrix
A diagonal matrix \( D \) can be represented as:
\[ D = \begin{pmatrix} d_1 & 0 & 0 & \cdots & 0 \\ 0 & d_2 & 0 & \cdots & 0 \\ 0 & 0 & d_3 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & d_n \end{pmatrix} \]

### Step 2: Determine the inverse of a diagonal matrix
The inverse of a diagonal matrix is also a diagonal matrix where each diagonal element is the reciprocal of the corresponding diagonal element of the original matrix. Therefore, the inverse \( D^{-1} \) can be expressed as:
\[ D^{-1} = \text{diag}\left(\frac{1}{d_1}, \frac{1}{d_2}, \frac{1}{d_3}, \ldots, \frac{1}{d_n}\right) \]

### Step 3: Write the inverse matrix explicitly
Thus, we can write:
\[ D^{-1} = \begin{pmatrix} \frac{1}{d_1} & 0 & 0 & \cdots & 0 \\ 0 & \frac{1}{d_2} & 0 & \cdots & 0 \\ 0 & 0 & \frac{1}{d_3} & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & \frac{1}{d_n} \end{pmatrix} \]

### Conclusion
Therefore, the inverse of the diagonal matrix \( D \) is given by:
\[ D^{-1} = \text{diag}\left(\frac{1}{d_1}, \frac{1}{d_2}, \frac{1}{d_3}, \ldots, \frac{1}{d_n}\right) \]