if A is an n x n matrix, then the number obtained by multiplying the entries in any row or column of A by corresponding cofactors and adding the resulting products is called the determinant of A, the sums themselves are called cofactor expansions of A

the best strategy for cofactor expansion is to expand along a row or a column with the most zeros

if A is an nxn triangular matrix (upper triangle matrix, lower triangle matrix, diagonal) then det(A) is the product of the entries on the main diagonal of the matrix

the arrow technique can work with 4x4 or higher matrices in finding the determinant

if A has a row of zeros or a column of zeros then the det(A) = 0

if A is a square matrix then det(A) .. det(A)T

In matrix B the first row of A was multiplied by k then

In the matrix B the first and second rows of A were interchanged

in the matrix B a multiple of the second row of A was added to the first row

if A is a square matrix with 2 proportional rows or 2 proportional columns then det(A) = …

det(kA) =

det(A+B) = det(A) + det(B)

det(A.B) = det(A). det(B) [Same Size]

A invertible then det(A)^-1 = 1/det(A)

The transpose of the cofactors matrix is named

A is invertible then A^-1 = adj(A)/det(A)

in Cramer’s law x1 = det(A)/det(A1)

A is nxn matrix then Ax = 0 has only the trivial solution

A is nxn matrix then the reduced row-echelon form of A is I

if A is a squared matrix then Ax = b is consistent for every nx1 matrix b

if A is a squared matrix then Ax = b has only one solution for every nx1 matrix b