Question

If abc ≠ 0, find the inverse of $\left[\begin{array}{ccc}a&0&0\\0&b&0\\0&0&c\end{array}\right]$

Answer 1

Answer:

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

Step-by-step explanation:

As we know that inverse of a matrix is calculated by Adjoint of matrix.

As

$A^{-1} =\frac{adj.A}{|A|}$

and adjoint of matrix A is calculated as

$adj.A=[A_{ji}]_{n\times n}$

so here

$A=\left[\begin{array}{ccc}a&0&0\\0&b&0\\0&0&c\end{array}\right]\\\\\\adj.A=\left[\begin{array}{ccc}bc&0&0\\0&ac&0\\0&0&ab\end{array}\right] ^{'}\\\\\\adj.A=\left[\begin{array}{ccc}bc&0&0\\0&ac&0\\0&0&ab\end{array}\right]$

Now

$|A|=\left |\begin{array}{ccc}a&0&0\\0&b&0\\0&0&c\end{array}\right |=abc \\\\\\A^{-1} =\frac{1}{abc}\left[\begin{array}{ccc}bc&0&0\\0&ac&0\\0&0&ab\end{array}\right]$

$A^{-1}=\left[\begin{array}{ccc}\frac{bc}{abc}&0&0\\0&\frac{ac}{abc}&0\\0&0&\frac{ab}{abc}\end{array}\right]\\\\\\A^{-1}=\left[\begin{array}{ccc}\frac{1}{a}&0&0\\0&\frac{1}{b}&0\\0&0&\frac{1}{c}\end{array}\right]$