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Is the determinant defined recursively?
The determinant can be defined recursively, where the determinant of a 1 by 1 matrix is the single entry in that matrix. Given a larger matrix, delete the first row and the jth column. If the general 2×2 matrix is written (a,b|c,d), its determinant is a×det(d)-b×det(c), or ad-bc.
Is determinant commutative?
As discussed below, the determinant is also defined for matrices whose entries are elements in more abstract algebraic structures known as commutative rings. The determinant can also be characterized as the unique function depending on the entries of the matrix satisfying certain properties.
What does a determinant tell you?
The determinant of a square matrix is a single number that, among other things, can be related to the area or volume of a region. In particular, the determinant of a matrix reflects how the linear transformation associated with the matrix can scale or reflect objects.
What does a determinant of 0 mean?
When the determinant of a matrix is zero, the volume of the region with sides given by its columns or rows is zero, which means the matrix considered as a transformation takes the basis vectors into vectors that are linearly dependent and define 0 volume.
What is determinant used for?
The determinant is useful for solving linear equations, capturing how linear transformation change area or volume, and changing variables in integrals. The determinant can be viewed as a function whose input is a square matrix and whose output is a number.
What does it mean when the determinant is 0?
Is determinant of a matrix commutative?
I know in general matrix multiplication is not commutative unless the matrices involved are diagonal and of the same dimension. However the determinant operator seems to not preserve the non commutative property of matrix multiplication, on either side of the equality.