A = {{-6,-2,1},{5,3,-2},{0,-2,1}}

Det[A]

A//MatrixForm

AA ={{3-λ ,0,0},{0,-4-λ,0},{0,0,-1-λ}}

Solve[Det[AA]==0,λ]

A = {{5,1,2,-9,8},{0,1,0,1,0},{1,-1,0,1,1},{1,0,1,0,1},{3,1,-6,-2,4}}

A//MatrixForm

Det[A]

Drop[A,{1},{1}]//MatrixForm

A[[1,1]]*Det[Drop[A,{1},{1}]]-A[[1,2]]*Det[Drop[A,{1},{2}]]+
A[[1,3]]*Det[Drop[A,{1},{3}]]-
A[[1,4]]*Det[Drop[A,{1},{4}]]+
A[[1,5]]*Det[Drop[A,{1},{5}]]

Sum[(-1)^(1+i)*A[[1,i]]*Det[Drop[A,{1},{i}]],{i,1,5}]

Det[A]

A//MatrixForm

(-1)^(3+2)*Det[Drop[A,{3},{2}]]

(Subscript[c, ##])

(CoMatrix = Array[(Subscript[c, ##])&, {5,5}])//MatrixForm

For[i=1,i≤ 5,i=i+1,
For[j=1, j≤ 5, j=j+1,
CoMatrix[[i,j]] = (-1)^(i+j)*Det[Drop[A,{i},{j}]];
]
]

For[j=1,j≤ 5,j=j+1,
For[i=1, i≤ 5, i=i+1,
CoMatrix[[i,j]] = (-1)^(i+j)*Det[Drop[A,{i},{j}]];
]
]

CoMatrix//MatrixForm

AdjMatrix = Transpose[CoMatrix]

AdjMatrix//MatrixForm

(Ainv= 1/Det[A]*AdjMatrix)//MatrixForm

A.Ainv//MatrixForm

Det[A]*Inverse[A]//MatrixForm

A\[Transpose]//MatrixForm

A//MatrixForm

(Subscript[(A+A\[Transpose]), i,j]) = (Subscript[A, i,j])+(Subscript[(A\[Transpose]), i,j]) = 
(Subscript[A, i,j])+(Subscript[A, j,i]) = (Subscript[A\[Transpose], j,i])+(Subscript[A, j,i]) = (Subscript[A, j,i])+(Subscript[A\[Transpose], j,i]) = (Subscript[(A+A\[Transpose]), j,i])