Question

If $A=\left[\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right]$ is a non-singular matrix, then show that $A$ is invertible and $A^{-1}=\frac{AdjA}{detA}$.

Answer 1

We know $A\left[adj\right(A\left)\right]$ gives diagonal matrix with $|A|$ on it's diagonal elements.

So $A\left[adj\right(A\left)\right]=|A|I$

$A^{-1}A\left[adj\right(A\left)\right]=A^{-1}|A|I$

$adj\left(A\right)=|A|A^{-1}$

So $A^{-1}=\frac{adj\left(A\right)}{|A|}$