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Determinants and operations on matrices

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Definition of determinant of a matrix as a product of the diagonal entries in a non-scaled echelon form.
The determinant of a matrix can be expressed as a product of the diagonal entries in a non-scaled echelon form.
Definition of the determinant in terms of the effect of elementary row operations
The permutation expansion for determinants
A matrix and its transpose have the same determinant.
Determinants multiply: det(AB) = det(A)det(B)
The determinant of the inverse of A is the reciprocal of the determinant of A.
The determinant of a block diagonal matrix is the product of the determinants of the blocks.

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The determinant of the inverse of A is the reciprocal of the determinant of A.

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Determinants multiply: det(AB) = det(A)det(B)Close match
A matrix and its transpose have the same determinant.Close match
Determinants and operations on matricesClose match
InverseClose match
Matrix transpose commutes with matrix inverse.Close match
The inverse of a matrix (if it exists) can be found by row reducing the matrix augmented by the identity matrix.Close match
Example of finding the inverse of a 2-by-2 matrix by using a formulaClose match
Matrix inverse is an involution.Close match
A determinant is a multilinear functionClose match
The eigenvalues of the inverse of a nonsingular matrix are the reciprocals of the eigenvalues.Close match
The inverse of a scalar multiple is the reciprocal times the inverse.Close match
Matrix inverses are unique: if A and B are square matricesClose match
Definition of the determinant in terms of the effect of elementary row operationsClose match
The inverse of a matrix can be expressed in terms of its matrix of cofactors.Close match
The determinant of a matrix can be expressed as a product of the diagonal entries in a non-scaled echelon form.Close match
For n-by-n invertible matrices A and BClose match
The inverse of a matrix can be used to solve a linear system.Close match
The determinant of a matrix can be computed as a cofactor expansion across any row.Close match
Determinants axiomaticallyClose match
DeterminantsClose match

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