1. Tardigrade
  2. Question
  3. Mathematics
  4. Statement 1: If A = [a&0&0 0&b&0 0&0&c] then A-1 = [(1/a)&0&0 0&(1/b)&0 0&0&(1/c)] Statement 2: The inverse of a diagonal matrix is a diagonal matrix.

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Q. Statement 1 : If $A = \begin{bmatrix}a&0&0\\ 0&b&0\\ 0&0&c\end{bmatrix} $ then $A^{-1} = \begin{bmatrix}\frac{1}{a}&0&0\\ 0&\frac{1}{b}&0\\ 0&0&\frac{1}{c}\end{bmatrix} $
Statement 2 : The inverse of a diagonal matrix is a diagonal matrix.

Matrices

Solution:

$A^{-1} = \frac{1}{\det A} \text{adj} A$
$ = \frac{1}{abc } \begin{bmatrix}bc&0&0\\ 0&ca&0\\ 0&0&ab\end{bmatrix}$
$= \begin{bmatrix}\frac{1}{a}&0&0\\ 0&\frac{1}{b}&0\\ 0&0&\frac{1}{c}\end{bmatrix} $
The inverse of a diagonal matrix is a diagonal matrix. Both true but Statement 2 is not correct reason of Statement 1.