1. For a square matrix, the sum of the product of elements of any row (or column) with their cofactors is always equal to determinants of the matrix.
2. For a square matrix, the sum of the product of elements of any row (or column) with the cofactors of corresponding elements of some other row (or column) is zero.
3. The value of a determinant remains unchanged if its rows and columns are interchanged.
4. If any two rows (or columns) of a determinant are interchanged, then the value of the determinant changes by minus sign only.
5. If any two rows or columns of a determinant are identical then its value is zero.
6. If each element of a row (or a column) of a determinant is multiplied by a constant k, then the value of the new determinant is ‘k’ times the value of the original determinant.
7. If each element of a row (or a column) of a determinant is expressed as a sum of two or more terms, then the determinant can be expressed as the sum of two or more determinants of the same order.
8. If each element of a row (or a column) of a determinant is multiplied by the same constant and then added to the corresponding elements of some other row (column) then the value of the determinant remains same.
9. If each element of a row (or column) in a determinant is zero, then its value is zero.
10. If the matrix is a diagonal square matrix then its determinant is the product of all the diagonal elements.
11. If A and B are square matrices of the same order, then
|AB| = |A| |B|
12. If a matrix is a triangular matrix of order n, then its determinant is the product of all the diagonal elements.
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