So i have this matrix
A= $$ \begin{pmatrix} 1 & 3 & 0 \\ -2 & -5 & 2 \\ 1 & 4 & 3 \\ \end{pmatrix} $$
And i want to find the inverse of it. Following all the calculations, i get that the determinant is 1 and that the adjucate of the matrix by creating the matrix of co factors is
$$ \begin{pmatrix} -23 & -9 & 6 \\ 8 & 3 & -2 \\ -3 & -1 & 1 \\ \end{pmatrix} $$ I checked the answer on the answer sheet and it's right.
And i would be happy to end the excercise here BUT if i remember correctly shouldn't i apply the sign rule?
I always thought that after having calculated the matrix, i apply the sign rule which would be
$$ \begin{pmatrix} + & - & + \\ - & + & - \\ + & - & + \\ \end{pmatrix} $$
If i apply the sign rule, then my matrix would look different:
$$ \begin{pmatrix} -23 & 9 & 6 \\ -8 & 3 & 2 \\ -3 & 1 & 1 \\ \end{pmatrix} $$
and ,in order to finish the calculations, i would need to multiply the adjucate by 1 over the determinant of A which is 1.
So i would multiply 1 times each entry in the adjugate and the matrix would still be
$$ \begin{pmatrix} -23 & 9 & 6 \\ -8 & 3 & 2 \\ -3 & 1 & 1 \\ \end{pmatrix} $$
Which, as you can clearly see, is not the correct answer.
So my question is , should i apply the sign rule? Should i not? What is it that i'm doing wrong? SHould i stop when i just find the adjugate and multiply the entries by the determinant ?
Thanks for the help!
(please do not tell me to find the inverse another way , thanks)!
You’ve already applied the sign rule when you calculated the cofactors.
For example when you calculate the $(1,1)$ element, you do:
$$M_{1,1}=\det\begin{pmatrix}-5&2\\ 4&3\end{pmatrix}=-23$$
And when you calculate the $(1,2)$ element, you do (note the negative sign!!):
$$M_{1,2}=-\det\begin{pmatrix}-2&2\\ 1&3\end{pmatrix}=-((-2*3)-(1*2))=8$$
Then to get the adjugate you take the transpose of the cofactors.
You’ve done everything correctly and incorporated the sign rule. No need to apply it twice.