If `D=diag [2, 3, 4]`, then `D^(-1)=`

Options

A. O
B. I
C. D
D. diag `[(1)/(2), (1)/(3), (1)/(4)]`

Answer

To find the inverse of the diagonal matrix \( D = \text{diag}[2, 3, 4] \), we can follow these steps:

### Step 1: Understand the Structure of the Diagonal Matrix
The diagonal matrix \( D \) can be represented as:
\[ D = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{pmatrix} \]

### Step 2: Calculate the Determinant of \( D \)
For a diagonal matrix, the determinant is the product of its diagonal elements:
\[ \text{det}(D) = 2 \times 3 \times 4 = 24 \]

### Step 3: Check if the Inverse Exists
Since the determinant \( \text{det}(D) \) is not equal to zero, the inverse of \( D \) exists.

### Step 4: Find the Inverse of the Diagonal Matrix
The inverse of a diagonal matrix is obtained by taking the reciprocal of each of its diagonal elements:
\[ D^{-1} = \text{diag}\left[\frac{1}{2}, \frac{1}{3}, \frac{1}{4}\right] \]
This can be represented as:
\[ D^{-1} = \begin{pmatrix} \frac{1}{2} & 0 & 0 \\ 0 & \frac{1}{3} & 0 \\ 0 & 0 & \frac{1}{4} \end{pmatrix} \]

### Final Result
Thus, the inverse of the matrix \( D \) is:
\[ D^{-1} = \begin{pmatrix} \frac{1}{2} & 0 & 0 \\ 0 & \frac{1}{3} & 0 \\ 0 & 0 & \frac{1}{4} \end{pmatrix} \]