What is minor expansion method?

What is minor expansion method?

Also known as “Laplacian” determinant expansion by minors, expansion by minors is a technique for computing the determinant of a given square matrix. . Although efficient for small matrices, techniques such as Gaussian elimination are much more efficient when the matrix size becomes large.

When can you use Laplace expansion?

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.

When would you use cofactor expansion?

Examples. Cofactor expansion can be very handy when the matrix has many 0’s. Let A=[1a0n−1B] where a is 1×(n−1), B is (n−1)×(n−1), and 0n−1 is an (n−1)-tuple of 0’s. Using the formula for expanding along column 1, we obtain just one term since Ai,1=0 for all i≥2.

What is expanding in determinants?

In linear algebra, the Laplace expansion, named after Pierre-Simon Laplace, also called cofactor expansion, is an expression of the determinant of an n × n matrix B as a weighted sum of minors, which are the determinants of some (n − 1) × (n − 1) submatrices of B.

How do you expand a column in determinants?

Expanding to Find the Determinant

1. Pick any row or column in the matrix. It does not matter which row or which column you use, the answer will be the same for any row.
2. Multiply every element in that row or column by its cofactor and add. The result is the determinant.

What is expansion by minors (cofactor expansion)?

It is computed by continuously breaking matrices down into smaller matrices until the 2×2 form is reached in a process called Expansion by Minors also known as Cofactor Expansion. We first define the minor matrix of as the matrix which is derived from by eliminating the row and column.

What is the determinant expansion by minors in matrices?

Determinant Expansion by Minors Also known as “Laplacian” determinant expansion by minors, expansion by minors is a technique for computing the determinant of a given square matrix. Although efficient for small matrices, techniques such as Gaussian elimination are much more efficient when the matrix size becomes large.

When to use matrix expansion in a matrix?

This can be especially useful if expansion about a particular row or column results in value for one or more of the cofactors since this eliminates an entire minor matrix from the calculation. Excercise 3-7.

How to write the determinant by expansion along row formally?

Taking these definitions together we can write the determinant by expansion along row formally as follows: It is valid to expand about any row or column. This can be especially useful if expansion about a particular row or column results in value for one or more of the cofactors since this eliminates an entire minor matrix from the calculation.