determinants

Summary

Determinant of 1x1 matrices

Determinant of 2x2 matrices

Determinant of 3x3 matrices, by cofactor expansion with the 1st row

Rule of Sarrus for 3x3 matrices

Rules

Adjoint formula

Equivalent statements of invertibility

if the determinant is 0, the matrix transforms(squishes) space into a lower dimension -> the inverse would not have a unique solution

Concept

Determinant

• visualise it as the degree to which a unit of space is scaled when undergoing a matrix transformation

Cofactor expansion

first minor, the submatrix derived by deleting the ith row and jth col
choose the row/col that has the most zeros

Reduction, with determinant of elementary matrices

• the detereminant of a matrix in RREF is easier to find

Adjoint

• (i,j) entry is the (j,i) cofactor

Cramer’s rule to solve linear systems

Application

Determinant of triangular matrices, product of the diagonal entries

Adjoint on a 2x2 matrix

Deriving the adjoint formular for inverse

Extra

Octave

# Determinant of a matrix
det(A)

# Adjugate matrix
adjoint(A)