Applications of Matrices and Determinants

The adjoint (or adjugate) of

The adjoint (or adjugate) of a square matrix A is defined as the transpose of the matrix of cofactors. For a 3×3 matrix, if we replace each element aᵢ

A square matrix A has

A square matrix A has an inverse A⁻¹ if and only if it is non-singular (|A| ≠ 0). The inverse is given by: A⁻¹ = (1/|A|) × adj A. This inverse satisfi

Elementary transformations include

Elementary transformations include: (1) Interchanging any two rows/columns (Rᵢ ↔ Rⱼ), (2) Multiplying a row/column by a non-zero scalar (Rᵢ → λRᵢ), an

A matrix is in row

A matrix is in row-echelon form when: (1) all zero rows appear below non-zero rows, (2) the first non-zero element (pivot) in each row is to the right

By the Rouché

By the Rouché-Capelli theorem, a system AX = B is consistent (has at least one solution) if and only if rank(A) = rank([A|B]), where [A|B] is the augm