Question

If $a,b,c$ are non-zero real numbers, then the inverse of the matrix
$A=\left[\begin{array}{ccc}a& 0& 0\\ 0& b& 0\\ 0& 0& c\end{array}\right]$
is equal to

• A. $\left[\begin{array}{ccc}{a}^{-1}& 0& 0\\ 0& b^{-1}& 0\\ 0& 0& c^{-1}\end{array}\right]$
• B. $\frac{1}{abc}\left[\begin{array}{ccc}{a}^{-1}& 0& 0\\ 0& b^{-1}& 0\\ 0& 0& c^{-1}\end{array}\right]$
• C. $\frac{1}{abc}\left[\begin{array}{ccc}{1}^{}& 0& 0\\ 0& 1^{}& 0\\ 0& 0& 1^{}\end{array}\right]$
• D. $\frac{1}{abc}\left[\begin{array}{ccc}{a}^{}& 0& 0\\ 0& b^{}& 0\\ 0& 0& c^{}\end{array}\right]$

Answer 1

Answer: (A)

$\left[\begin{array}{ccc}{a}^{-1}& 0& 0\\ 0& b^{-1}& 0\\ 0& 0& c^{-1}\end{array}\right]$

Given $A=\left[\begin{array}{ccc}a& 0& 0\\ 0& b& 0\\ 0& 0& c\end{array}\right]$,

We know that $A^{-1}=\frac{adjA}{\left|A\right|}$

$⟹adjA={\left[\begin{array}{ccc}bc& 0& 0\\ 0& ac& 0\\ 0& 0& ab\end{array}\right]}^T=\left[\begin{array}{ccc}bc& 0& 0\\ 0& ac& 0\\ 0& 0& ab\end{array}\right]$ and $\left|A\right|=abc$

$⟹A^{-1}=\left[\begin{array}{ccc}{a}^{-1}& 0& 0\\ 0& b^{-1}& 0\\ 0& 0& c^{-1}\end{array}\right]$