Question

If \(D=\text{diag}(d_1,d_2,\ldots,d_n)\), where \(d_i\neq 0\), for \(i=1,2,\ldots,n\), then \(D^{-1}\) is equal to

• A. \(D^T\)
• B. \(D\)
• C. \(\text{Adj}(D)\)
• D. \(\text{diag}(d_1^{-1},d_2^{-1},\ldots,d_n^{-1})\)

Hint:

Inverse of a diagonal matrix is obtained by taking reciprocal of each diagonal entry (if all are nonzero).

Answer 1

The Correct Option is D

Step 1: Recall definition of diagonal matrix.
A diagonal matrix has nonzero elements only on the diagonal:
\[ D=\text{diag}(d_1,d_2,\ldots,d_n) \]
Step 2: Condition for inverse.
A diagonal matrix is invertible iff all diagonal entries are nonzero.
Given \(d_i\neq 0\), so inverse exists.
Step 3: Inverse of diagonal matrix.
To get \(D^{-1}\), each diagonal element becomes its reciprocal:
\[ D^{-1}=\text{diag}\left(\frac{1}{d_1},\frac{1}{d_2},\ldots,\frac{1}{d_n}\right) \]
Step 4: Verification.
\[ DD^{-1}=\text{diag}(d_1\cdot d_1^{-1},\ldots,d_n\cdot d_n^{-1})=I \]
Final Answer:
\[ \boxed{\text{diag}(d_1^{-1},d_2^{-1},\ldots,d_n^{-1})} \]