Answer to Question #258645 in Linear Algebra for Tege

Question

Find the cofactor for given matrix $\begin{vmatrix} 2& 5\\ 7 & -5 \end{vmatrix}$

Answer 1

$M_{1,1}=(-1)^{1+1}\cdot (-5)=-5;M_{1,2}=(-1)^{1+2}\cdot 7=-7;\\ M_{2,1}=(-1)^{2+1}\cdot 5=-5; M_{2,2}=(-1)^{2+2}\cdot 2=2;\\$

Cofactor(A)=$\begin{pmatrix} M_{1,1} & M_{1,2}\\ M_{2,1} & M_{2,2} \end{pmatrix}$ =$\begin{pmatrix} -5 & -7\\ -5 & 2 \end{pmatrix}$

More explanations:

$\begin{vmatrix} \colorbox{aqua}2 & \colorbox{aqua}5\\ \colorbox{aqua} 7 & -5 \end{vmatrix}$ $M_{1,1}=(-1)^{1+1}\cdot (-5)=-5$

$\begin{vmatrix} \colorbox{aqua}2 & \colorbox{aqua} 5\\ 7 &\colorbox{aqua} -5 \end{vmatrix}$ $M_{1,2}=(-1)^{1+2}\cdot 7=-7$

$\begin{vmatrix} \colorbox{aqua}2 & 5\\ \colorbox{aqua}7 &\colorbox{aqua} -5 \end{vmatrix}$ $M_{2,1}=(-1)^{2+1}\cdot 5=-5$

$\begin{vmatrix} 2 & \colorbox{aqua}5\\ \colorbox{aqua}7 &\colorbox{aqua} -5 \end{vmatrix}$ $M_{2,2}=(-1)^{2+2}\cdot 2=2;$