Question

if A= |a1 b1 c1| |a2 b2 c2| |a3 b3 c3| is a non singular matrix and A is invertable prove thar A inverse = AdjA/detA​

Answer 1

Step-by-step explanation:

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Answer 2

Given:

A is a non-singular matrix.

To find:

The nonsingular matrix and A is invertible prove that A inverse=$\frac{adjA}{detA}$

Step-by-step explanation:

$A=\left[\begin{array}{ccc}a_{1}&b_{1}&c_{1}\\a_{2}&b_{2}&c_{2}\\a_{3}&b_{3}&c_{3}\end{array}\right]$

$A[adjA]=\left[\begin{array}{ccc}a_{1}&b_{1}&c_{1}\\a_{2}&b_{2}&c_{2}\\a_{3}&b_{3}&c_{3}\end{array}\right] \left[\begin{array}{ccc}c_{11}&c_{21}&c_{31}\\c_{12}&c_{22}&c_{32}\\c_{13}&c_{23}&c_{33}\end{array}\right]$

$A[adjA]=\left[\begin{array}{ccc}detA&0&0\\0&detA&0\\0&0&detA\end{array}\right]$

because A is invertible, $detA\neq 0,$

$\frac{1}{detA}A(adjA)=I$

$A[\frac{1}{detA}adjA]=I$

$\frac{1}{detA}adjA=A^{-}$

$A^{-}=\frac{adjA}{detA}$

Answer:

Therefore, the A is invertible prove that $A^{-}=\frac{adjA}{detA}$.