Determinant of any square matrix is equal to determinant of its transpose. Lets take an example of any square matrix and find value of its determinant. Then transpose this matrix and again find value of determinant of transpose of matrix. We will note that determinant of matrix is equal to determinant of its transpose.
How to prove that det(adj(A))= (det(A)) power n-1?
Note: This property holds for square matrices which are invertible.
This property of adjoint of matrices can be easily proved using property
where adj(A) is adjoint of A, det(A) is determinant of A and
is inverse of A. A here is an invertible matrix.
Value of Determinant remains unchanged if we add equal multiples of all the elements of row (column) to corresponding elements of another row (column)
If, we have a given matrix A. We can perform some column and row operations on this matrix such that value of its determinant remains unchanged. If, we multiply particular row (column) of matrix A with a constant k and then add all the element of that row (column) to the corresponding elements of another row (column) then the value of determinant remains unchanged. This is an important property regarding matrices and determinants.
What is the determinant of a matrix if all the elements in a row or column are zero?
If in a given matrix, we have all zero elements in a particular row or column then determinant of such a matrix is equal to zero.