What is expanding the determinant?

What is expanding the determinant?

When calculating the determinant, you can choose to expand any row or any column. Regardless of your choice, you will always get the same number which is the determinant of the matrix A. This method of evaluating a determinant by expanding along a row or a column is called Laplace Expansion or Cofactor Expansion.

What is determinant theorem?

Two of the most important theorems about determinants are yet to be proved: Theorem 1: If A and B are both n × n matrices, then detAdetB = det(AB). Theorem 2: A square matrix is invertible if and only if its determinant is non-zero.

How do you find the determinant of cofactor expansion?

One way of computing the determinant of an n×n matrix A is to use the following formula called the cofactor formula. Pick any i∈{1,…,n}. Then det(A)=(−1)i+1Ai,1det(A(i∣1))+(−1)i+2Ai,2det(A(i∣2))+⋯+(−1)i+nAi,ndet(A(i∣n)).

What is Laplace expansion theorem?

The Laplace expansion is a formula that allows us to express the determinant of a matrix as a linear combination of determinants of smaller matrices, called minors. The Laplace expansion also allows us to write the inverse of a matrix in terms of its signed minors, called cofactors.

Why is det AB )= det A det B?

If A and B are n × n matrices, then det(AB) = (detA)(detB). In other words, the determinant of a product of two matrices is just the product of the deter- minants.

What is determinant function?

In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It allows characterizing some properties of the matrix and the linear map represented by the matrix.

What is determinant of cofactor?

Cofactor of a Determinant The cofactor is defined as the signed minor. Cofactor of an element aij, denoted by Aij is defined by A = (–1)i+j M, where M is minor of aij.

Why is determinant used?

Determinants can be used to give explicit formulas for the solution of a system of n equations in n unknowns, and for the inverse of an invertible matrix. They can also be used to give formulas for the area/volume of certain geometric figures.