Question

Se da matricea A= (m+2 m m)
(m m+2 m)
(m m m+2)
m apartine R.
Sa se determine m stiind ca inversa matricei A este egala cu adjuncta matricei A.

Answer 1

$\begin{vmatrix}m+2 & m & m \\ m & m+2 & m \\ m & m & m+2\notag\end{vmatrix} =\begin{vmatrix}3m+2 & 3m+2 & 3m+2 \\ m & m+2 & m \\ m & m & m+2\notag\end{vmatrix} = \\\\\\(3m+2)\begin{vmatrix}1 & 1 & 1 \\ m & m+2 & m \\ m & m & m+2\notag\end{vmatrix} = (3m+2)\begin{vmatrix}1 & 0 & 0 \\ m & 2 & 0 \\ m & 0 & 2\notag\end{vmatrix} =\\ \\=(3m+2)(4) = 12m+8$

$A^{-1} = \frac{1}{det(A)}*A^*\\ A^{-1}=A^* \implies det(A)=1\\12m+8=1\\12m=-7\\m=\frac{-7}{12}$