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|