Finding element of inverse matrix

[Yankel]

Hello all,

I have this matrix A

\[A=\begin{pmatrix} 1 &2 &3 &4 \\ 9 &8 &2 &0 \\ 17 &2 &0 &0 \\ 1 &0 &0 &0 \end{pmatrix}\]

B is defined as the inverse of A. I need to find the element in the first row and fourth column of B, without using determinants, so without using adjoint.

How should I do it then ?

Thanks !

 

[Deveno]

We know that the dot-product of the first 3 rows of $A$ with the fourth column of $B$ is 0, and the dot product of the fourth row of $A$ with the fourth column of $B$ is 1.

let's call this column vector $(b_1,b_2,b_3,b_4)$. We need to find $b_1$. From the last sentence of my previous paragraph, we know that:

$1b_1 + 0b_2 + 0b_3 + 0b_4 = 1$.

Your conclusion?

 

[Yankel]

I didn't see this. Nice one, thanks !