##### Lecture on Properties of Determinants
Content covered:
• Important properties of determinant are:
1. The value of Δ does not change if rows and columns are interchanged.
|AT|=|A|
$\begin{vmatrix}2&3\\4&5\end{vmatrix}$ = $\begin{vmatrix}2&4\\3 &5\end{vmatrix}$
2. If any row or column of a matrix is 0 then |A|=0
Δ of
$\begin{vmatrix}0&0&0&0\\7&5&9&6\\5&3&2&1\\7&6&2&5\end{vmatrix}$ =0
3. If matrix have identical rows or column . The value of Δ =0.
Δ of
$\begin{vmatrix}6&5&3\\2&9&7\\6&5&3\end{vmatrix}$ =0
4. The value of a Δ is unchanged if row or column are added m times.
A= $\begin{vmatrix}7&6\\2 &1\end{vmatrix} =-5$
After R1<-R1+2R2
$\begin{vmatrix}11&8\\2 &7\end{vmatrix} =-5$
5. Multiplication with scalar
A=$\begin{vmatrix}a1&a2\\ a3 &a4\end{vmatrix}$
A'= $\begin{vmatrix}k a1&k a2\\ a3 &a4\end{vmatrix} =k|A|$
6. Product with cofactor
$\begin{vmatrix}a1&a2\\a3 &a4\end{vmatrix} =k^n|A|$ where n is order of Δ
a1*cf(a1)+a2*cf(a2)=|A|
a1*cf(a3)+a2*cf(a4)=0
7. Determinant of multiplication of two matrices =Determinant of first matrix X Determinant of second matrix
|AB|=|A| X |B|
9. If any two rows (or two columns )of a determinant are interchanged, the value of the determinant is multiplied by -1.
10. If all element of a row (or a column ) of a determinant are multiplied by the same number k, the value of the new determinant is k times of the given determinant.
11. If A be an n-rowed square matrix and 'k' be any scalar then                  |kA|=kn|A|
12. In a determinant the sum of the product of the element of any row(column) with the cofactors of the corresponding elements of any other row(column) is zero.                                                                                                                                                 Δ= ai1Ai1+ai2Ai2+--------------------------ainAin                                                                          but if ai1Aj1+ai2Aj2+--------------------------ainAjn​=0
13. In a determinant ,each element in any row (or column ) Consists of the sum of two terms,then the determinant can be expressed as sum of two determinant of the same order.                                                                                                +
14. The value of upper triangular or lower triangular matrices or diagonal matrix always be multiplication of diagonal element.
15. The value of the determinant of skew symmetric matrix of odd order is always zero.