Maths WorkshopMaths problems with full solutions

Determinants

34 problems · Properties of determinants, cofactors and Cramer's rule.

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Ex.231Determinant of a 2×2 matrix

#determinant #2x2 matrices #invertibility

Ex.449Determinant of a 3×3 matrix by Sarrus' rule

#determinant #Sarrus rule #3x3 matrices

Ex.450Determinant of an inverse

#inverse #determinant #formula

Ex.451Effect of row operations on the determinant

#row operations #determinant properties #multilinearity

Ex.452Determinant of a triangular matrix

#triangular matrix #determinant #diagonal

Ex.896The determinant is alternating

#alternating #equal columns #sign change

Ex.897Determinant of a block triangular matrix

#block matrices #determinant #triangular

Ex.898A trigonometric determinant

#trigonometric determinant #cos #sin #identity

Ex.899Cramer's rule for a 2×2 system

#Cramer's rule #linear system #determinant

Ex.900Determinant of a cyclic permutation matrix

#permutation matrix #cyclic #determinant

Ex.901Determinant of a diagonalisable matrix

#diagonalisation #determinant #eigenvalues

Ex.902Expansion along a column

#cofactor expansion #minors #determinant

Ex.903Expansion along a row

#cofactor expansion #minors #determinant

Ex.904Multilinearity of the determinant

#multilinearity #determinant properties #proof

Ex.905Determinant of an orthogonal matrix

#orthogonal matrix #determinant #isometry

Ex.906Determinant and rank

#determinant #rank #invertibility

Ex.907Determinant of the transpose

#transpose #determinant #proof

Ex.1631Computing the inverse with cofactors

#cofactor #comatrix #inverse

Ex.1632The cofactor formula for the inverse

#cofactor #comatrix #inverse formula

Ex.1633Determinant of a companion matrix

#companion matrix #determinant #characteristic polynomial

Ex.1634Cramer's rule for a 3×3 system

#Cramer's rule #3x3 determinant #linear system

Ex.1635A cyclic 3×3 determinant

#cyclic determinant #factorisation #roots of unity

Ex.1636A mixed determinant exercise

#determinant #row operations #factorisation

Ex.1637The determinant as a volume

#determinant #volume #parallelepiped

Ex.1638A numerical 4×4 determinant

#4x4 determinant #row reduction #computation

Ex.1639Invertibility depending on a parameter

#parameter #invertibility #determinant

Ex.1640Determinant of a product

#determinant of product #multiplicativity #proof

Ex.1641Determinant of a skew-symmetric matrix

#skew-symmetric #determinant #odd dimension

Ex.1642Trace and determinant via eigenvalues

#characteristic polynomial #trace #determinant #eigenvalues

Ex.1643The Vandermonde determinant

#Vandermonde #determinant #factorisation

Ex.2118Gram determinant and Cauchy-Schwarz

#Gram determinant #Cauchy-Schwarz #inner product

Ex.2119The general Vandermonde determinant

#Vandermonde #determinant #general formula

Ex.2120Hadamard's inequality

#Hadamard inequality #determinant #column norms

Ex.2121Sylvester's determinant identity

#Sylvester identity #block matrices #determinant