Inverse of Diagonal Matrix
The inverse of a diagonal matrix is a special case of finding the inverse of a matrix. To find the inverse of diagonal matrix, we use a formula and do not require to find the determinant and adjoint the diagonal matrix. Let us first recall the meaning of a diagonal matrix which is a square matrix with nonzero elements on the main diagonal and the remaining nondiagonal elements are zero. The inverse of a diagonal matrix can be determined by replacing diagonal elements of the matrix with their reciprocals.
In this article, we will explore the concept of finding the inverse of a diagonal matrix, its formula and proof. We will also go through the necessary condition required to find the inverse of the diagonal matrix and solve a few examples based on the concept for a better understanding of the topic.
1.  What is Inverse of Diagonal Matrix? 
2.  Inverse of Diagonal Matrix Formula 
3.  Inverse of Diagonal Matrix Proof 
4.  Inverse of Diagonal Matrix Theorem 
5.  FAQs on Inverse of Diagonal Matrix 
What is Inverse of Diagonal Matrix?
The inverse of a diagonal matrix is given by replacing the main diagonal elements of the matrix with their reciprocals. The inverse of the diagonal matrix exists if and only if all elements of the main diagonal are nonzero and this is a necessary condition for the inverse of a diagonal matrix to exist. As a diagonal matrix is a square matrix, therefore its inverse is also a square matrix and exists if the diagonal matrix is nonsingular, i.e., has a nonzero determinant. Now, let us see the formula to find the inverse of a diagonal matrix.
Inverse of Diagonal Matrix Formula
To generalize the formula for the inverse of diagonal matrix, let us see the specific cases first. First, we will write the formula to find the inverse of 2 × 2 diagonal matrix. For a diagonal matrix A of order 2, we have
\(A = \left[\begin{array}{ccc} a_{11} & 0 \\ \\
0 & a_{22}
\end{array}\right] \), then its inverse is
\(A^{1} = \left[\begin{array}{ccc} \frac{1}{a_{11}} & 0 \\ \\
0 & \frac{1}{a_{22}}
\end{array}\right] \)
Now, to find the formula of the inverse of a diagonal matrix of order, we have
If \(A = \left[\begin{array}{ccc}
a_{11} & 0 & 0 \\
0 & a_{22} & 0 \\
0 & 0 & a_{33}
\end{array}\right] \), then its inverse is given by,
\(A^{1} = \left[\begin{array}{ccc}
\frac{1}{a_{11}} & 0 & 0 \\
0 & \frac{1}{a_{22}} & 0 \\
0 & 0 & \frac{1}{a_{33}}
\end{array}\right] \)
Now, the image below shows the general formula for the inverse of diagonal matrix of order n:
Inverse of Diagonal Matrix Proof
Consider a diagonal matrix of order 2 given by \(A = \left[\begin{array}{ccc} a & 0 \\ \\
0 & b
\end{array}\right] \)
Now, assume the inverse of diagonal matrix A to be \(A^{1} = \left[\begin{array}{ccc} p & q \\ \\
r & s
\end{array}\right] \)
Using the definition of inverse of a matrix, we have
\(\left[\begin{array}{ccc} 1 & 0 \\ \\
0 & 1
\end{array}\right] =I = AA^{1} = \left[\begin{array}{ccc} a & 0 \\ \\
0 & b
\end{array}\right] \left[\begin{array}{ccc} p & q \\ \\
r & s
\end{array}\right] = \left[\begin{array}{ccc} ap & aq \\ \\
br & bs
\end{array}\right]\)
Therefore, we have
1 = ap ⇒ p = 1/a
0 = aq ⇒ q = 0
1 = bs ⇒ s = 1/b
0 = br ⇒ r = 0
Using the above values, we have the inverse of diagonal matrix A as,
\(A^{1} = \left[\begin{array}{ccc} \frac{1}{a} & 0 \\ \\
0 & \frac{1}{b}
\end{array}\right] \)
As we can see, the inverse of diagonal matrix A is a diagonal matrix with main diagonal elements as reciprocals of diagonal elements of matrix A. Now, for a diagonal matrix D of order n, we have
\(DD^{1} = \begin{bmatrix}
d_{11}&0&\cdots &0 \\
0&d_{22}&\cdots &0 \\
\vdots & \vdots & \ddots & \vdots\\
0&0&\cdots &d_{nn}
\end{bmatrix}\) \(\begin{bmatrix}
\frac{1}{d_{11}}&0&\cdots &0 \\
0&\frac{1}{d_{22}}&\cdots &0 \\
\vdots & \vdots & \ddots & \vdots\\
0&0&\cdots &\frac{1}{d_{nn}}
\end{bmatrix}\) \(=\begin{bmatrix}
1&0&\cdots &0 \\
0&1&\cdots &0 \\
\vdots & \vdots & \ddots & \vdots\\
0&0&\cdots &1
\end{bmatrix}\)
Inverse of Diagonal Matrix Theorem
Statement: The theorem on the inverse of diagonal matrix states that a diagonal matrix D = diag(d_{1}, d_{2}, d_{3}, ..., d_{n}) is invertible if and only if all diagonal entries are nonzero, i.e., d_{i} ≠ 0 for 1 ≤ i ≤ n.
Proof: For a diagonal matrix D = diag(d_{1}, d_{2}, d_{3}, ..., d_{n}), if all d_{i} ≠ 0, then
diag(d_{1}, d_{2}, d_{3}, ..., d_{n}) diag(1/d_{1}, 1/d_{2}, 1/d_{3}, ..., 1/d_{n}) = diag(1, 1, 1, ..., 1) = I and diag(1/d_{1}, 1/d_{2}, 1/d_{3}, ..., 1/d_{n}) diag(d_{1}, d_{2}, d_{3}, ..., d_{n}) = diag(1, 1, 1, ..., 1) = I
Now, assume d_{i} = 0 for some i, then the i^{th} row of the matrix D becomes zero, and hence the matrix becomes noninvertible.
Important Notes on Inverse of Diagonal Matrix
 The inverse of a diagonal matrix is a diagonal matrix where the elements of the main diagonal are the reciprocals of the corresponding elements of the original diagonal matrix.
 If any element of the main diagonal of a diagonal matrix is zero, then the entire row of the matrix becomes zero, which makes the matrix singular, and hence, the inverse of the diagonal matrix won't exist.
 The necessary condition of the inverse of the diagonal matrix to exist is that all main diagonal elements must be nonzero.
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Inverse of Diagonal Matrix Examples

Example 1: Find the inverse of diagonal matrix \(B = \left[\begin{array}{ccc} 2 & 0 \\ \\
0 & 7
\end{array}\right] \)Solution: Using the formula for the inverse of diagonal matrix, the diagonal elements of B^{1} are 1/2 and 1/7.
Answer: Therefore, \(B^{1} = \left[\begin{array}{ccc} 1/2 & 0 \\ \\
0 & 1/7
\end{array}\right] \) 
Example 2: Check if the inverse of the matrix \(D = \left[\begin{array}{ccc} 2 & 0 \\ \\
0 & 0
\end{array}\right] \) exists.Solution: As we can see, row 2 of matrix D is equal to 0, this implies the matrix is singular and hence, has a determinant equal to 0.
Although, all nondiagonal elements of the matrix D are zero which implies it is a diagonal matrix.
Therefore, matrix D is a diagonal matrix but it is not invertible as all main diagonal are not nonzero.
Answer: Inverse of diagonal matrix D does not exist.

Example 3: Find the inverse of the diagonal matrix A = \(\left[\begin{array}{rrr}
2 & 0 & 0 \\
0 & 3 & 0 \\
0 & 0 & 5
\end{array}\right]\).Solution:
The inverse of A is found by replacing each of its diagonal element by its reciprocal. Then:
A^{1 }= \(\left[\begin{array}{rrr}
1/2 & 0 & 0 \\
0 & 1/3 & 0 \\
0 & 0 & 1/5
\end{array}\right]\)Answer: The inverse of the given diagonal matrix is, \(\left[\begin{array}{rrr}
1/2 & 0 & 0 \\
0 & 1/3 & 0 \\
0 & 0 & 1/5
\end{array}\right]\).
FAQs on Inverse of Diagonal Matrix
What is Inverse of Diagonal Matrix in Algebra?
The inverse of a diagonal matrix is given by replacing the main diagonal elements of the matrix with their reciprocals. The inverse of a diagonal matrix is a special case of finding the inverse of a matrix.
What is the Formula for Inverse of Diagonal Matrix?
For a diagonal matrix D = diag(d_{1}, d_{2}, d_{3}, ..., d_{n}) with d_{i} ≠ 0 for 1 ≤ i ≤ n, the inverse of diagonal matrix D is D^{1} = diag(1/d_{1}, 1/d_{2}, 1/d_{3}, ..., 1/d_{n})
When Does the Inverse of Diagonal Matrix Exist?
The inverse of a diagonal matrix exists if and only if all main diagonal elements of the matrix are nonzero.
How to Find Inverse of Diagonal Matrix?
To find the inverse of diagonal matrix, we use a formula and do not require to find the determinant and adjoint the diagonal matrix. We replace the main diagonal elements of the matrix with their reciprocals to find its inverse.
Is the Inverse of Diagonal Matrix a Diagonal Matrix?
Yes, the inverse of diagonal matrix is a diagonal matrix.
Is the Inverse of Diagonal Matrix a Symmetric Matrix?
Yes, the inverse of diagonal matrix is a symmetric matrix.
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