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.
Determinant of Matrix becomes k times by multiplying any row or column by k
If, we have any given matrix A and we multiply any row or column of matrix with constant k then determinant of modified matrix becomes k times of determinant of A. We can show this by taking example of 3 x 3 matrix A and calculating its determinant.
Determinant of a matrix changes sign if we interchange any two rows or columns present in a matrix
Determinant of a matrix changes its sign if we interchange any two rows or columns present in a matrix. We can prove this property by taking an example. We take matrix A and we calculate its determinant (|A|).
Determinant of Matrix is equal to Determinant of its Transpose
If, we have any given matrix A then determinant of matrix A is equal to determinant of its transpose. We can prove this by taking variable
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.
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