Inverse of Diagonal Matrices

# The Inverse of a Diagonal Matrix

Suppose that we have the following $n \times n$ diagonal matrix $D = \begin{bmatrix} d_1 & 0 & \cdots &0 \\ 0 & d_2 & 0 & 0\\ \vdots & 0 & \ddots & \vdots \\ 0 & 0 & \cdots & d_n \end{bmatrix}$. This diagonal matrix is invertible if all of the entries on the main diagonal are nonzero, that is for every $i$, $d_i ≠ 0$. The inverse of matrix $D$ will also be a diagonal $n \times n$ matrix in the following form:

(1)
\begin{align} D^{-1} = \begin{bmatrix}\frac{1}{d_1} & 0 & \cdots &0 \\ 0 & \frac{1}{d_2} & 0 & 0\\ \vdots & 0 & \ddots & \vdots \\ 0 & 0 & \cdots & \frac{1}{d_n}\end{bmatrix} \end{align}

Therefore, to form the inverse of a diagonal matrix, we will take the reciprocals of the entries in the main diagonal. For example, consider the following diagonal matrix $A = \begin{bmatrix} 2 & 0 & 0\\ 0 & 3 & 0\\ 0 & 0 & 4 \end{bmatrix}$. Taking the reciprocals of the main diagonal, we obtain that $A^{-1} = \begin{bmatrix} \frac{1}{2} & 0 & 0\\ 0 & \frac{1}{3} & 0\\ 0 & 0 & \frac{1}{4} \end{bmatrix}$.