# determinant

(4.8 hours to learn)

## Summary

The determinant is a scalar value associated with a square matrix. It is convenient algebraically because it behaves nicely with respect to matrix multiplication, inverses, and transposes, as well as the Gaussian elimination operations. It gives the factor by which volumes are rescaled by the matrix's associated linear transformation. It also equals the product of the eigenvalues.

## Context

This concept has the prerequisites:

## Goals

• Define the determinant (in terms of a sum over column permutations)
• Be able to calculate the determinant recursively in terms of cofactors
• Know what the Gaussian elimination operations do to the determinant of a matrix
• Be able to manipulate the determinant algebraically, together with matrix multiplication, inverse, and transpose
• Derive the identities relating the determinant to matrix multiplication, inverse, and transpose
• Show that a matrix is invertible if and only if its determinant is nonzero
• Show that the determinant of a triangular matrix is the product of the diagonal entries

## -Free-

MIT Open Courseware: Linear Algebra (2011)
Videos for an introductory linear algebra course focusing on numerical methods.
Author: Gilbert Strang