Summary On Properties of Determinants
- Let $A =(a_{ij})_{nxn}$ be a diagonal ot triangular matrix, then $det(A)$ is the product of diagonal entries.
- Interchanging rows or columns of a square matrix changes only the sign of its determinant.
- Adding constant multiple of a row or column of a square matrix A on to an-other row or column of A doesn’t change its determinant.
- Multiplying a row or column of a square matrix A by any constant k its determinant equals k times det(A).
- If A is a square matrix of order n and k is a scalar then $det(kA) = k^{n} det(A)$
- If A is a square matrix of order n and k is a scalar then $det(kA) = k^{n} det(A)$
- Determinant of a square matrix A and determinant of its transpose is the same.
- Let A and B be two square matrices of the same order , then $det(AB) = det(A)det(B)$
- For any square matrix A: $det(A^{m} = (det(A))^{m}$, for $m \in \mathbb{Z^{+}}$
- .Let A be an invertible square matrix, then $det(A^{-1} = \frac{1}{det(A)}$
